EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6409
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Construction and Classification of Generalized
Hadamard Codes over Eisenstein Local Rings Z2s[ω]

Muhammad Sajjad1,∗, Muhammad Farhan Ali Khan2, Maha Alammari3,
Robinson-Julian Serna4

1 NUTECH School of Applied Science and Humanities, National University of Technology,
Islamabad, 44000, Pakistan
2 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
3 Department of Mathematics, College of Science, King Saud University, P.O. Box 22452
Riyadh 11495, Saudi Arabia
4 Escuela de Matemáticas y Estad́ıstica, Universidad Pedagógica y Tecnológica de Colombia,
Tunja, Colombia

Abstract. The research paper examines the design principles and structural features of General-
ized Hadamard (GH) codes that operate within Eisenstein local rings Z2s [ω], utilizing a primitive
cube root of unity ω that satisfies the relation ω2 + ω + 1 = 0. The paper first introduces an al-
gebraic Eisenstein integer framework before developing an appropriate Gray mapping to examine
binary-domain representations of these codes. We establish the essential criteria and necessary
checks for determining the linear properties of GH codes based on Z2s [ω] structures. This research
defines the kernel structure of these codes together with their rank specification and an evaluation
of their structural properties. A classification system for Z2s [ω]-linear Hadamard codes is pre-
sented in the final part of the paper, based on their algebraic and combinatorial characteristics.
Future studies on coding techniques within algebraic integer rings can build upon this work, as
our research expands the understanding of code theory over non-traditional rings.

2020 Mathematics Subject Classifications: 94B75, 11T71, 94A24, 68P30, 14G50, 94A05

Key Words and Phrases: Generalized Hadamard Codes, Eisenstein Integers, Local Rings, Gray
Map, Code Linearity, Code Kernel

1. Introduction

Modulator–Demodulator, as we are all accustomed to it, is one of the cornerstones of
contemporary digital communication, thanks to which we are able to detect and correct
errors in transmitted data. It is mainly a theory for the construction of structured codes
that can efficiently handle errors while remaining data-intuitive. It laid the foundations

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6409

Email addresses: muhammad.sajjad@nutech.edu.pk (M. Sajjad), muhammadfarhan20117@gmail.com (M.
F. A. Khan), malammari@ksu.edu.sa (M. Alammari), robinson.serna@uptc.edu.co (R. J. Serna)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 2 of 32

for several critical concepts in this area, including linear codes, cyclic codes, and BCH
codes, which have seen practical applications in data storage, satellite communication,
and cybersecurity [1–4]. The use of algebraic structures, such as rings and fields, to define
codes with robust or stable properties has led to the development of more general and
efficient error-correcting techniques [5–7]. These advances were complemented by Gray
maps, isometries, and duality concepts, linking binary code representations with algebraic
tools [7–9]. In recent years, this algebraic framework has been extended to non-field
structures, such as finite rings, group rings, and number-theoretic rings like Gaussian and
Eisenstein integers [4, 10–12].

Many algebraic domains have studied Hadamard codes—a class of codes with a simple
structure and an optimal minimum distance. These codes, originally built from Hadamard
matrices, are a backbone of many applications due to their strong error detection and sim-
ple structure. Researchers have explored the rank and kernel properties in the binary and
Z4-linear settings to understand whether the algebraic complexity of these problems im-
pacts code dimension. The study of Hadamard codes over Z2s , Zps , and mixed modules
has allowed a deeper examination of structural invariants and equivalence classes of such
codes [4, 7, 9]. These studies highlight the influence of the underlying ring on linearity, de-
coding, and code equivalence, motivating extensions to code constructions over Eisenstein
and quaternion integers [9, 13, 14].

Recently, much effort has been directed toward classifying and constructing generalized
Hadamard (GH) codes over various algebraic structures. Bhunia et al. [15, 16] developed
a framework for Zps-linear GH codes in terms of kernel, linearity, and equivalence. This
builds upon previous work by Dougherty, Villanueva, and Rifà [6, 17, 18], and other
contributions [5, 19, 20] that focused on rank and kernel of codes over Z2s and related
rings. These investigations have greatly enhanced the understanding of code structure,
Gray maps, and their connection to classification theory. The significance of Z4-linear
codes was demonstrated by Carlet [5] and Hammons et al. [7], who revealed that Z4-
linear codes underpin other well-known nonlinear codes such as Kerdock and Preparata
codes. Further studies into other ring-based codes include extended perfect codes and
duality over Z2k , as explored by Krotov [21, 22]. More recent work by Shi et al. [9, 23] has
focused on additive codes over mixed rings, duality principles, and classification criteria,
demonstrating the algebraic depth and practical relevance of such constructions.

The other significant direction has been the exploration of Hadamard and generalized
Hadamard (GH) codes over number-theoretic rings. In particular, Sajjad et al. have made
substantial contributions to coding theory in the context of Gaussian and Eisenstein inte-
gers [10, 12], including BCH code constructions and alternant codes with applications in
modern technology [24]. Modified Berlekamp–Massey algorithms, along with included al-
gebraic tools, have been employed in their decoding frameworks, which generalize classical
coding theory into broader algebraic domains [4, 11]. Additionally, the work of Villanueva
and Zinoviev [25, 26] on Hadamard matrix construction has influenced generalized code
design across diverse metrics and algebraic rings.

Despite the fact that the theory of GH codes over classical rings like Z2s is fairly well
established, there exists a significant gap in the literature concerning their generalization



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 3 of 32

over Eisenstein integers and corresponding rational domains. Eisenstein integers offer a
rich algebraic structure with complex arithmetic, which has already proven useful in BCH
and alternant code design. It is anticipated that the promising results of Sajjad et al.
[10, 12, 24] in robust code construction and error correction using Eisenstein integers will
yield analogous benefits in the context of GH codes. Furthermore, intellectual stimulation
arises from the complexity of such a non-trivial ring system, where understanding linearity,
the behaviour of the Gray map, and the structure of the kernel becomes crucial. As the
foundational works on ring-based and non-binary Hadamard codes [6, 15, 17, 27] already
offer natural starting points for generalization, the Eisenstein local ring Z2s [ω], where
ω2 + ω + 1 = 0, is chosen as the natural candidate for extension.

Finally, this study helps to bridge a critical gap between classical GH code theory
and a novel algebraic domain, with potential applications in secure communications and
high-reliability systems.

This article contributes the following:

• It demonstrates how GH codes can be constructed over Eisenstein local rings Z2s [ω],
where ω2 + ω + 1 = 0.

• It defines and analyzes a Gray map suitable for Eisenstein local rings with binary
image representation.

• It obtains linearity conditions for GH codes with respect to elements in Z2s [ω].

• It investigates the kernel and rank structures of these codes, leading to the determi-
nation of some of their algebraic invariants.

• It partially classifies Z2s [ω]-linear Hadamard codes in terms of their structural and
combinatorial properties.

• It distinguishes classical GH code theory from modern algebraic settings such as
Eisenstein rings.

2. Eisenstein Integers [12, 24]

Let ω = −1+i
√
3

2 be a primitive cube root of unity. Then the identity

1 + ω + ω2 = 0

implies that ω2 = −ω − 1.
The set of Eisenstein integers consists of complex numbers that can be written in the

form a+ ωb, where a, b ∈ Z. Mathematically, this set is defined as

E = {a+ ωb | a, b ∈ Z}.

The set E forms a ring.



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 4 of 32

The conjugate of z = a+ ωb ∈ E is given by

z∗ = a+ ω2b.

For example, z∗ = 1 + 2ω2 is the conjugate of z = a+ bω in the set E.
Every Eisenstein integer z has a norm denoted by N(z), which is defined as

N(z) = zz∗ = a2 − ab+ b2.

Moreover, the norm is multiplicative:

N(z1z2) = N(z1)N(z2), for all z1, z2 ∈ E.

Proposition 2.1 [12, 24]: For ω = −1+i
√
3

2 , the ring E is a Euclidean domain.
The units in the ring E of Eisenstein integers are ±1,±ω,±ω2.
The primes in E include:

• Rational primes p satisfying p ≡ 2 (mod 3),

• Eisenstein integers z such that N(z) = p, where p is a prime.

The quotient ring E/nE is canonically isomorphic to the ring

En = {a+ bω | a, b ∈ Zn},

which is the ring of Eisenstein integers modulo n. This ring is also a principal ideal ring.
Lemma 2.1 [24]: Let z = a+ ωb ∈ En. Then z is a unit in En if and only if N(z) is

a unit in Zn.
Since our main focus is on local rings of Eisenstein integers, we consider n = ps, where

p is a prime integer and s is a positive integer.
Note that Eps is not always a local ring, unlike Zps . Similarly, it Ep is not always a

field even when p is prime.
Theorem 2.1 [12, 24]: For p ≥ 3, the ring Eps is local if and only if p ≡ 2 (mod 3)

or p = 2.

3. Gray Map and Related Results [6, 16, 17]

In this section, we provide the definition of the generalized Gray map for Z2s [ω] GH
codes. Then we establish some properties of the Gray map for Z2s [ω], based on the results
given in Section 2 of [6, 16, 17].

Let ϕs be Carlet’s Gray map from Z2s [ω] to Z22(s−1)

2 [ω], defined as

ϕs(h) = (hs−1, hs−1, . . . , hs−1) + (h0, . . . , hs−2)Ys−1,

where h ∈ Z2s [ω], and [h0, h1, . . . , hs−1]2 is the binary (2-ary) expansion of h, i.e.,

h =
s−1∑
i=0

2ihi,



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 5 of 32

with hi ∈ Z2[ω]. Here, Ys−1 is an (s − 1) × 22(s−1) matrix whose columns are all distinct
elements of Zs−1

2 [ω].
Now we extend the map ϕs to a component-wise map:

φs : Zn
2s [ω] −→ Zn·22(s−1)

2 [ω].

The matrix Ys−1 can be obtained recursively, starting from

Y1 =
[
00 01 10 11

]
,

and for s > 1,

Ys =

(
Ys−1 Ys−1

00 01
Ys−1 Ys−1

10 11

)
.

Example 3.1. Let s = 2. Take h ∈ Z4[ω], where h = h0 + 2h1 is the 2-ary expansion of
h. Then ϕ2 is a Gray map from Z4[ω] to Z4

2[ω], which is given in Table 1. As h ∈ Z4[ω]
with binary representation [h0, h1]2. The Carlet’s generalized Gray map ϕ2 is defined as:

ϕ2(h) = (h1, h1, h1, h1) + h0Y1 ∈ Z4
2[ω],

where Y1 = (00, 01, 10, 11).

Table 1: Gray map from Z4[ω] to Z4
2[ω]

h ∈ Z4[ω] [h0, h1]2 ϕ2(h) h ∈ Z4[ω] [h0, h1]2 ϕ2(h)

00 [00, 00]2 (00, 00, 00, 00) 20 [00, 10]2 (10, 10, 10, 10)
01 [01, 00]2 (00, 11, 01, 10) 21 [01, 10]2 (10, 01, 11, 00)
02 [00, 01]2 (01, 01, 01, 01) 22 [00, 11]2 (11, 11, 11, 11)
03 [01, 01]2 (01, 10, 00, 11) 23 [01, 11]2 (11, 00, 10, 01)
10 [10, 00]2 (00, 01, 10, 11) 30 [10, 10]2 (10, 11, 00, 01)
11 [11, 00]2 (00, 10, 11, 01) 31 [11, 10]2 (10, 00, 01, 11)
12 [10, 01]2 (01, 00, 11, 10) 32 [10, 11]2 (11, 10, 01, 00)
13 [11, 01]2 (01, 11, 10, 00) 33 [11, 11]2 (11, 01, 00, 10)

Now some results for the above-defined Carlet’s generalized Gray map are presented
below.

Let ek be the vector that has 1 in the kth position and 0 elsewhere. Let u, v ∈ Z2s [ω]
and [u0, u1, . . . , us−1]2, [v0, v1, . . . , vs−1]2 be the 2-ary expansions of u and v, respectively,
i.e.,

u =
s−1∑
i=0

2iui, v =
s−1∑
i=0

2ivi.

Now define the operation ⊕2 for elements of Z2s [ω] as:



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 6 of 32

u⊕2 v =

s−1∑
i=0

ri2
i, where ri = ui + vi mod 2 in Z2[ω],

and the operation ⊙2 as:

u⊙2 v =

s−1∑
i=0

ti2
i,

where

ti =

{
10 if ui + vi ≥ 2,

00 otherwise.

The 2-ary expansion of u⊙2 v is [t0, t1, . . . , ts−1]2, where ti ∈ {00, 10}.
Lemma 3.1: Let u ∈ Z2s [ω] and µ ∈ Z2[ω]. Then

ϕs(u+ µ2s−1) = ϕs(u) + (µ, µ, . . . , µ).

Proof: Since

u+ µ2s−1 = u1 + µ02
s−1 + µ2s−1 = u1 + (µ0 + µ)2s−1,

where Let u1 ∈
{
00, . . . , 0 · 2s−1 − 1, . . . , 2s−1 − 1 · 0, . . . , 2s−1 − 1 · 2s−1 − 1

}
and u0 ∈

Z2[ω]. Then, by the definition of the Gray map ϕs, we have:

ϕs(u+ µ2s−1) = ϕs(u1) + (µ0 + µ, . . . , µ0 + µ)

= ϕs(u1) + (µ0, . . . , µ0) + (µ, . . . , µ)

= ϕs(u1) + (µ, . . . , µ).

Corollary 3.1: Let λ, µ ∈ Z2[ω]. Then,

ϕs(λµ2
s−1) = λϕs(µ2

s−1) = λµϕs(2
s−1).

Proof: By the definition of the Gray map ϕs, we have ϕs(µ2
s−1) = (µ, . . . , µ). Then,

ϕs(λµ2
s−1) = (λµ, . . . , λµ) = λ(µ, . . . , µ) = λϕs(µ2

s−1) = λµϕs(2
s−1).

Proposition 3.1: Let u, v ∈ Z2s [ω]. Then

ϕs(u) + ϕs(v) = ϕs(u⊕2 v).

Proof: Let [u0, u1, . . . , us−1]2 and [v0, v1, . . . , vs−1]2 be the 2-ary expansions of u and
v, respectively. Let yi be the (i+ 1)-th row of Y , for 0 ≤ i ≤ s− 2. Then,

ϕs(u) = (us−1, us−1, . . . , us−1) +

s−2∑
i=0

uiyi,



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 7 of 32

ϕs(v) = (vs−1, vs−1, . . . , vs−1) +
s−2∑
i=0

viyi.

Therefore,

ϕs(u) + ϕs(v) = (rs−1, rs−1, . . . , rs−1) +
s−2∑
i=0

riyi = ϕs(u⊕2 v),

where ri = ui + vi in Z2[ω] for 0 ≤ i ≤ s− 1.

Proposition 3.2: Let u, v ∈ Z2s [ω]. Then

u⊕2 v = u+ v − 2(u⊙2 v).

Proof: Let [u0, u1, . . . , us−1]2, [v0, v1, . . . , vs−1]2 be the 2-ary expansions of u and v,
respectively. Note that 0 ≤ ui + vi ≤ 2. By the division algorithm,

ui + vi = 2ti + ri,

where ti = 1 if ui + vi ≥ 2, and ti = 0 otherwise; also 0 ≤ ri ≤ 1. Then we have:

u+ v =
s−1∑
i=0

(ui + vi)2
i

=

s−1∑
i=0

(2ti + ri)2
i

= 2

s−1∑
i=0

ti2
i +

s−1∑
i=0

ri2
i

= 2(u⊙2 v) + u⊕2 v.

Therefore,
u⊕2 v = u+ v − 2(u⊙2 v).

Corollary 3.2: Let u, v ∈ Z2s [ω]. Then,

ϕs(u) + ϕs(v) = ϕs(u+ v − 2(u⊙2 v)).

Proof: From Proposition 3.1, ϕs(u) + ϕs(v) = ϕs(u ⊕2 v). From Proposition 3.2,
u⊕2 v = u+ v − 2(u⊙2 v). So,

ϕs(u) + ϕs(v) = ϕs(u⊕2 v) = ϕs(u+ v − 2(u⊙2 v)).

Corollary 3.3: Let u, v ∈ Z2s [ω]. Then,

2s−1(u⊕2 v) = 2s−1(u+ v).



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 8 of 32

Proof: Let [u0, u1, . . . , us−1]2, [v0, v1, . . . , vs−1]2 be the binary expansions of u and v,
respectively. Then [0, 0, . . . , 0, u0]2 is the binary expansion of 2s−1u, so 2s−1u⊙2 v is 2s−1

if u0 + vs−1 ≥ 2, and 0 otherwise. In any case, 2(2s−1u⊙2 v) = 0. Hence, by Proposition
3.2, the result follows.

Corollary 3.4: Let u ∈ Z2s [ω] and [u0, u1, . . . , us−1]2 be its binary expansion. Then,
for any i ∈ {0, . . . , s− 2},

ϕs(u) + ϕs(2
i) = ϕs(u+ 2i − 2i+1ti),

where

ti =

{
1 if ui ≥ 1,

0 otherwise.

Corollary 3.5: Let v ∈ Z2s [ω]. Then,

ϕs(2
s−1 + v) = ϕs(2

s−1) + ϕs(v).

Corollary 3.6: Let u, v ∈ Z2s [ω]. Then,

ϕs(2
s−1u+ v) = ϕs(2

s−1u) + ϕs(v).

Lemma 3.2: Let u ∈ {(01)2s−2, (03)2s−2, . . . , (31)2s−1, (33)2s−1} ⊂ Z2s [ω]. Then,

ϕs(u) + ϕs(0 · 2s−2) = ϕs(u+ 0 · 2s−2 + 0 · 2s−1).

Corollary 3.7: Let v ∈ {(01)2s−2, (03)2s−2, (21)2s−2, (23)2s−2} and U = {(01)2s−2, (03)2s−2, . . . , (31)2s−1, (33)2s−1} ⊂
Z2s [ω]. Then,

ϕs(u) + ϕs(v) =

{
ϕs(u+ v + 0 · 2s−1) if u ∈ U,

ϕs(u+ v) if u ∈ Z2s [ω] \ U.

Lemma 3.3: Let u ∈ {(10)2s−2, . . . , (13)2s−2, (30)2s−1, . . . , (33)2s−1} ⊂ Z2s [ω]. Then,

ϕs(u) + ϕs(2
s−2 · 0) = ϕs(u+ 2s−2 · 0 + 2s−1 · 0).

Corollary 3.8: Let v ∈ {(10)2s−2, (12)2s−2, (30)2s−2, (32)2s−2} and let

U ′ = {(10)2s−2, . . . , (13)2s−2, (30)2s−1, . . . , (33)2s−1} ⊂ Z2s [ω].

Then,

ϕs(u) + ϕs(v) =

{
ϕs(u+ v + 2s−1 · 0) if u ∈ U ′,

ϕs(u+ v) if u ∈ Z2s [ω] \ U ′.

Corollary 3.9: Let v ∈ {(11)2s−2, (13)2s−2, (31)2s−2, (33)2s−2}, and define

U1 = {(01)2s−2, (03)2s−2, (21)2s−1, (23)2s−1},
U2 = {(10)2s−2, (12)2s−2, (30)2s−1, (32)2s−1},
U3 = {(11)2s−2, (13)2s−2, (31)2s−1, (33)2s−1}.



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 9 of 32

Then,

ϕs(u) + ϕs(v) =


ϕs(u+ v + 02s−1) if u ∈ U1,

ϕs(u+ v + 2s−1 · 0) if u ∈ U2,

ϕs(u+ v + 2s−1 · 2s−1) if u ∈ U3,

ϕs(u+ v) if u ∈ Z2s [ω] \ (U1 ∪ U2 ∪ U3).

Lemma 3.4: Let µk ∈ Z2[ω], k ∈ {0, . . . , s− 2}. Then,

s−2∑
k=0

µkϕs(2
k) = ϕs

(
s−2∑
k=0

µk2
k

)
, where 2k ∈ Z2s [ω].

Proof: Let yk be the k-th row of matrix Y . By definition, we have

s−2∑
k=0

µkϕs(2
k) =

s−2∑
k=0

µkek+1Y =
s−2∑
k=0

µkyk+1 = µY,

where µ = (µ0, . . . , µs−2). Since [µ0, . . . , µs−2, 0] is the binary expansion of
∑s−2

k=0 µk2
k,

we conclude that

µY = ϕs

(
s−2∑
k=0

µk2
k

)
.

Proposition 3.3. Let u, v ∈ Z2s [ω]. Then,

ϕs(u) + ϕs(v) = ϕs(u− v) = (µ, . . . , µ) if u− v = µ2s−1 ∈ 2s−1Z2s [ω] \ {00},

and

ϕs(u)−ϕs(v) contains each element of Z2[ω] exactly 22(s−2) times if u−v ∈ Z2s [ω]\2s−1Z2s [ω].

Proof: If u−v = λ2s−1 ∈ 2s−1Z2s [ω]\{0}, then by Lemma 3.1, ϕs(u) = ϕs(v)+(λ, . . . , λ),
so

ϕs(u)− ϕs(v) = (λ, . . . , λ) = ϕs(λ2
s−1) = ϕs(u− v).

Now assume that u − v ∈ Z2s [ω] \ 2s−1Z2s [ω]. Without loss of generality, either u ∈
2s−1Z2s [ω], v ∈ Z2s [ω] \ 2s−1Z2s [ω] or u, v ∈ Z2s [ω] \ 2s−1Z2s [ω].

For the first case, ϕs(u) = (λ1, . . . , λ1) and ϕs(v) = ϕs(v1) + (λ2, . . . , λ2), where v1 ∈
{01, . . . , 2s−1 − 1, . . . , (2s−1 − 1)(2s−1 − 1)}, λ1, λ2 ∈ Z2[ω]. Note that ϕs(v1) is a nonzero
row of the GH matrix H(22, 22(s−2)) corresponding to the GH code ϕs(Z2s [ω]). Therefore,
ϕs(v1) contains each element of Z2[ω] exactly 22(s−2) times and hence ϕs(u)−ϕs(v) contains
each element of Z2[ω] exactly 22(s−2) times.

For the second case, ϕs(u) = ϕs(u1) + (λ1, . . . , λ1) and ϕs(v) = ϕs(v1) + (λ2, . . . , λ2),
where u1, v1 ∈ {01, . . . , 2s−1 − 1, . . . , (2s−1 − 1)(2s−1 − 1)} and λ1, λ2 ∈ Z2[ω]. Note that
both ϕs(u1) and ϕs(v1) are nonzero rows of H(22, 22(s−2)), so they contain each element of



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 10 of 32

Z2[ω] exactly 22(s−2) times, and hence ϕs(u)−ϕs(v) contains each element of Z2[ω] exactly
22(s−2) times.

Proposition 3.4. Let u, v ∈ Z2s [ω]. Then,

dH(ϕs(u), ϕs(v)) = wtH(ϕs(u− v)).

Proof: If u = 0 or v = 0, the result is trivially true. Assume u ̸= 0 and v ̸= 0, and
consider three cases. First, if u = v, the result is trivially true.

Second, if u− v ∈ 2s−1Z2s [ω]\{0}, then by Proposition 3.3, ϕs(u)−ϕs(v) = ϕs(u− v),
and hence

dH(ϕs(u), ϕs(v)) = wtH(ϕs(u− v)).

Finally, assume that u, v ∈ Z2s [ω] \ 2s−1Z2s [ω]. By Proposition 3.3, ϕs(u) − ϕs(v)
contains each element of Z2[ω] exactly 22(s−2) times, and hence

dH(ϕs(u), ϕs(v)) = 3 · 22(s−2) = wtH(ϕs(u− v)).

4. Construction of GH Codes over Z2s [ω]

Let

Ti = {jk · 2i−1 : j, k ∈ {0, 1, . . . , 2s−i+1 − 1}} for all i ∈ {1, . . . , s}.

Note that
T1 = {00, 01, . . . , 2s − 1 2s−1 − 1}.

Let t1, t2, . . . , ts be nonnegative integers with t1 ≥ 1. Consider the matrix

A
(t1,...,ts)
2

whose columns are exactly all the vectors of the form zt, where

z ∈ {0} × T t1−1
1 × T t2

2 × · · · × T ts
s .

Let
00, 01, . . . , 2s − 1 2s − 1

be the vectors having the same element 00, 01, . . . , 2s − 1 from Z2s [ω] in all coordinates,
respectively.

Any matrix A
(t1,...,ts)
2 can also be obtained by applying the recursive construction given

below. Start with the matrix
A

(1,0,...,0)
2 = (1 0).

If we have a matrix A
(t1,...,ts)
2 , then for any i ∈ {1, . . . , s}, we can construct the matrix

Ai =

[
A A · · · A · · · A · · · A

2i−1 · 00 2i−1 · 01 · · · 2i−1 · 0(2s−i+1 − 1) · · · 2i−1 · (2s−i+1 − 1)0 · · · 2i−1 · (2s−i+1 − 1)(2s−i+1 − 1)

]



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 11 of 32

Finally, by permuting the rows of Ai, we obtain a matrix

A
(t′1,...,t

′
s)

2

where t′j = tj for j ̸= i. Note that by permuting the columns of Ai, another matrix

A
(t′1,...,t

′
s)

2 can also be obtained.

To construct matrices recursively, starting from the base matrix A
(1,0,...,0)
2 , proceed in

the following way. First, to obtain matrix A
(t1,0,...,0)
2 , we add t1 − 1 rows of order 2s, then

t2 rows of order 2s−1, and so on, up to generate A
(t1,t2,...,0)
2 ; and finally we add ts rows of

order 2 to achieve
A

(t1,...,ts)
2 .

Let H̃
(t1,...,ts)
2 be the Z2s [ω]-additive code of type

(n, t1, . . . , ts)

generated by the matrix A
(t1,...,ts)
2 , where t1, t2, . . . , ts are nonnegative integers with t1 ≥ 1.

Let

n = 22(t−s+1), where t =

(
s∑

i=1

(s− i+ 1)ti

)
− 1.

The code H̃
(t1,...,ts)
2 has length n, and the corresponding Z2s [ω]-linear code

H
(t1,...,ts)
2 = φs

(
H̃

(t1,...,ts)
2

)
is a Generalized Hadamard code of length 22t.
Example 4.1: For s = 2, we have the following matrices which generate codes over Z4[ω].

A1,1 =

[
10 10 10 10
00 02 20 22

]

A1,2 =

 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
00 02 20 22 00 02 20 22 00 02 20 22 00 02 20 22
00 00 00 00 02 02 02 02 20 20 20 20 22 22 22 22



A2,0 =

[
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
00 01 02 03 10 11 12 13 20 21 22 23 30 31 32 33

]

Example 4.2: For s = 3, the following are generator matrices.

A1,0,1 =

[
10 10 10 10
00 04 40 44

]



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 12 of 32

A1,1,0 =

[
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
00 02 04 06 20 22 24 26 40 42 44 46 60 62 64 66

]
Example 4.3: The additive code H̄(1,0,...,0) is generated by

A
(1,0,...,0)
2 = (10),

so H̄(1,0,...,0) = Z2s [ω]. This additive code has length n = 1, cardinality 22s, and minimum
distance 1. Thus,

H(1,0,...,0) = φs(H̄(1,0,...,0))

has length N = 22(s−1), cardinality 4N = 22s, and minimum distance 3. Since

N

4
= 3 · 22(s−2),

it is clearly a binary generalized Hadamard code and also a linear code.

Example 4.4: For λ = 1, the normalized GH matrix is given as:

H(4, 1) =


00 00 00 00
00 01 10 11
00 11 01 10
00 10 11 01


Then,

FH = {(00, 00, 00, 00), (00, 01, 10, 11), (00, 11, 01, 10), (00, 10, 11, 01)} ,

and
CH =

⋃
α∈Z2[ω]

(FH + α · 10) .

Here, CH is a linear GH code over Z2[ω] of length 4, and

CH = H1,0 = φs(H̄1,0) = φs(Z4[ω]),

where H̄1,0 is generated by A1,0
2 = (10).

Theorem 4.1: Let t1, . . . , ts be non-negative integers with t1 ≥ 1. The Z2s [ω]-linear
code H(t1,...,ts) of type (n, t1, . . . , ts) is a generalized Hadamard (GH) code over Z2s [ω] of
length 22t, where

t =

(
s∑

i=1

(s− i+ 1) · ti

)
− 1 and n = 22(t−s+1).



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 13 of 32

Proof. Let H̄ = H̄(t1,...,ts) be the Z2s [ω]-additive code of length n. Consider the matrix

A = A
(t1,t2,...,ts)
2 as its generator. The additive code can be written as

H̄ =
⋃

λ∈Z2[ω]

(
AH̄ + λ · 2s−1

)
,

where

AH̄ = {h mod 2s−1 : h ∈ H̄} and AH̄ + λ · 2s−1 = {h+ λ · 2s−1 : h ∈ H̄}.

Then by Lemma 3.1,

H = φs(H̄) =
⋃

λ∈Z2[ω]

(φs(AH̄) + λ · 10) .

The code H has length 22t = n · 22(s−1) and cardinality 22(t+1) = n · 22s.
It is enough to show that φs(AH̄) is the set of rows of a generalized Hadamard matrix

H(22, 22(s−2)n).
Let us consider two distinct elements u, v ∈ AH̄. We need to show that φs(u)−φs(v)

contains each element of Z2[ω] exactly 22(s−2)n times.
We analyze two scenarios based on the order of u − v:
Case 1: If ord(u − v) = 2, then by the established construction, u − v includes all

elements of 2s−1 · Z2s [ω] exactly n/4 times. Therefore,

φs(u − v) includes all elements of Z2[ω] exactly
22(s−1) · n

4
= 22(s−2)n times.

By Proposition 3.3, φs(u − v) = φs(u) − φs(v), and hence φs(u) − φs(v) comprises
all elements of Z2[ω] exactly 22(s−2)n times.

Case 2: If ord(u − v) ≥ 2, then following the construction, u − v includes all ele-
ments of 2s−1 · Z2s [ω] exactly α times (α ≥ 0), and the remaining n− 4α coordinates are
from Z2s [ω] \ 2s−1 · Z2s [ω].

Then by Proposition 3.3, we have:

φs(u)−φs(v) includes all elements of Z2[ω] exactly α·22(s−1)+(n−4α)·22(s−2) = 22(s−2)n times.

Therefore, H is a generalized Hadamard code over Z2s [ω].

5. Linearity of Z2s [ω]-Linear GH Codes

This section establishes several results concerning the linearity of Z2s [ω]-linear GH
codes by generalizing the results given in Section 4.

Theorem 5.1: The Z2s [ω]-linear Hadamard codes H(1,0,...,0) and H(1,0,...,0,1,0), with
s > 2, are linear.

Proof: By Example 4.4, we know that H(1,0,...,0) is linear. Now, we examine H̃ =
H̃(1,0,...,0,1,0) and H = φs(H̃). Recall that the code H of length 16 is constructed from:



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 14 of 32

A
(1,0,...,0,1,0)
2 =

(
10 10 · · · 10 10
00 01 · 2s−2 · · · 32 · 2s−2 33 · 2s−2

)
Let αi = (2i 0, 2i 0, . . . , 2i 0, 2i 0) for 0 ≤ i ≤ s− 1,

αs =(00,0 2s−1,00,0 2s−1,2s−1 0,2s−1 2s−1,2s−1 0,2s−1 2s−1,00,0 2s−1,00,0 2s−1,2s−1 0,2s−1 2s−1,2s−1 0,2s−1 2s−1)

αs+1 = (00, 01 · 2s−2, . . . , 33 · 2s−2)

Suppose C represents the linear code constructed from

B = {φs(αi) : 0 ≤ i ≤ s+ 1}.

We now prove that C ⊆ H. Let c =
∑s+1

i=0 λiφs(αi) ∈ C, where λi ∈ Z2[ω].
By Corollary 3.5, it is sufficient to observe:

c′ = λs+1φs(αs+1) +
s−2∑
i=0

λiφs(αi) ∈ H.

If λs+1 = 0, then c′ ∈ H since

s−2∑
k=0

λiφs(αi) = φs

(
s−2∑
k=0

λiαi

)
.

If λs+1 = 10, then:

c′ = φs(00, 01 · 2s−2, . . . , 33 · 2s−2) + ϕs(u, u, . . . , u),

where u =
∑s−2

k=0 λi2
i 0.

Let us define:

U = {(01) · 2s−2, (03) · 2s−2, . . . , (31) · 2s−1, (33) · 2s−1}
U ′ = {(10) · 2s−2, . . . , (13) · 2s−2, (30) · 2s−1, . . . , (33) · 2s−1}
U1 = {(01) · 2s−2, (03) · 2s−2, (21) · 2s−1, (23) · 2s−1}
U2 = {(10) · 2s−2, (12) · 2s−2, (30) · 2s−1, (32) · 2s−1}
U3 = {(11) · 2s−2, (13) · 2s−2, (31) · 2s−1, (33) · 2s−1}

Then, by Corollaries 3.7, 3.8, and 3.9:

c′ = φs(00, 01 · 2s−2, . . . , 33 · 2s−2) + φs(u, u, . . . , u) + αs.

If u ∈ U ∪ U ′, then

c′ = φs(00, 01 · 2s−2, . . . , 33 · 2s−2) + ϕs(u, u, . . . , u),



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 15 of 32

otherwise. In both cases, c′ ∈ H.
For λs+1 = 01 and λs+1 = 11, it can be proven similarly using Corollaries 3.7, 3.8, and

3.9.
Since |C| = |H| = 22(s+2), it follows that C = H, and therefore, H exhibits linearity.
Theorem 5.2: The codes H(1,0,...,0,1,ts) and H(1,0,...,0,ts), with s > 2 and ts ≥ 0, are

the only Z2s [ω]-linear Hadamard codes that exhibit linearity.
Proof: Initially, we prove the linearity of these codes using induction on ts. By

Theorem 5.1, the codes H(1,0,...,0) and H(1,0,...,0,1,0) exhibit linearity.
We hypothesize that H = φs(H̄), where H̄ = H(1,0,...,0,ts−1,ts), ts−1 ∈ {0, 1} and ts ≥ 0,

is linear.
Now, we prove that Hs = H(1,0,...,0,ts−1,ts+1) is linear. Through iterative construction,

Hs =
{
φs

(
(h,h,h,h) + λ(00, (01)2s−1, (10)2s−1, (11)2s−1)

)
: h ∈ H̄, λ ∈ Z2[ω]

}
,

which simplifies to:{(
φs(h), φs(h+ λ(01)2s−1), φs(h+ λ(10)2s−1), φs(h+ λ(11)2s−1)

)
: h ∈ H̄, λ ∈ Z2[ω]

}
.

By Corollaries 3.1 and 3.6, this equals:{
(h′,h′ + λ · 01,h′ + λ · 10,h′ + λ · 11) : h′ ∈ H,λ ∈ Z2[ω]

}
.

Thus, it is possible to partition Hs into 4-blocks as follows:

Hs00 = {(h′,h′,h′,h′) : h′ ∈ H},
Hs01 = {(h′,h′ + 01 · 01, h′ + 01 · 10,h′ + 01 · 11) : h′ ∈ H},
Hs10 = {(h′,h′ + 10 · 01, h′ + 10 · 10,h′ + 10 · 11) : h′ ∈ H},
Hs11 = {(h′,h′ + 11 · 01, h′ + 11 · 10,h′ + 11 · 11) : h′ ∈ H}.

Given that H is linear, it is clear that the sum of any two vectors from Hs will lie in
one of the blocks Hs00, Hs01, Hs10, Hs11. Therefore, Hs is linear.

We now demonstrate the nonlinearity of H = φs(H̄), where H̄ = H(1,0,...,0,2,0). Let

r = (00,012s−2, . . . ,332s−2).

H has length 256 and is constructed from

A
(1,0,...,0,2,0)
2 =

10 10 10 · · · 10 10
r r r · · · r r
00 (01)2s−2 (02)2s−2 · · · (32) (33)2s−2

 .

By Corollaries 3.5, 3.7, 3.8, and 3.9, we have:

φs(r, r, . . . , r, r) + φs(00, (01)2
s−2, . . . , (33)2s−2) = φs(z),

where
z = (r, r, . . . , r) + (00, (01)2s−2, . . . , (33)2s−2) + p,



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 16 of 32

with
p = (0, u1, 0, u1, u2, u3, u2, u3, 0, u1, 0, u1, u2, u3, u2, u3),

and

u1 = (00, 2s−1, 00, 2s−1, . . . , 00, 2s−1),

u2 = (00, 00, 00, 00, 2s−10, . . . , 2s−10),

u3 = (00, 2s−1, 00, 2s−1, 2s−10, 2s−12s−1, . . . , 2s−12s−1).

Since φs(p) = 28 · 22(s−1) < 3N
4 , where N is the length of H, we have φs(p) /∈ H, and

thus φs(z) /∈ H. Therefore, H = H(1,0,...,0,2,0) is nonlinear.
Let H = φs(H̄), where H̄ = H(t1,...,ts). For any i ∈ {1, . . . , s}, define Hi = φs(H̄i),

where H̄i = H(t′1,...,t
′
s), t′i = ti + 1 and t′j = tj for j ̸= i.

We consider thatH = φs(H̄), where H̄ = H(1,0,...,0). Now, we establish the nonlinearity
of Hi for every i ∈ {1, . . . , s− 2}. The generator matrix of H̄i contains two nonzero rows:

w1 = 10, w2 = 2i−1(00, . . . , 0, 10, . . . , 12s+1−i, . . .).

Let w2j be the j-th coordinate of w2 and [(w2j)0, (w2j)1, . . . , (w2j)s−1]p its p-ary ex-
pansion. By Corollary 3.4,

ϕs(w2j) + ϕs(2
i−1) = ϕs(w2j + 2i−1 − zj),

where zj = 2i if (w2j)i−1 ≥ 1, and 0 otherwise. Then,

φs(w2) + φs(2
i−1) = φs(w2 + 2i−1 − z),

where z = (z1, z2, . . . , z22(s+1−i)) ∈ Z22(s+1−i)

2s and zj = 2i for even k ∈ {2, 4, . . . , 2s+1−i}
and zj = 0 otherwise.

We just need to show z /∈ H̄i. Note that wtH(φs(z)) = 22(s−i) · wtH(φs(2
i)). If

i ∈ {1, . . . , s− 2}, then
wtH(φs(z)) = 6 · 22(2s−i−2).

But the code Hi has minimum distance

3 · 22(2s−i−1) > wtH(φs(z)),

therefore, φs(z) /∈ Hi, for i ∈ {1, . . . , s− 2}.
Finally, in general, for H = φs(Ĥ), where Ĥ = Ĥ(t1,...,ts), we demonstrate that when-

ever H is nonlinear, Hi remains nonlinear for all i ∈ {1, . . . , s}.
Let us assume that Hi does not exhibit linearity. Then, by considering iterative con-

struction, for any u,v ∈ Ĥ, we have that

(u, . . . ,u), (v, . . . ,v) ∈ Ĥi.

Moreover, since Hi is linear,



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 17 of 32

φs(u, . . . ,u) + φs(v, . . . ,v)

= φs(a, . . . ,a)+λ·2i−1
(
0, . . . , 0, 2s−i+1−1, . . . , 2s−i+1−1, 0, . . . , 2s−i+1−1, 2s−i+1−1

)
∈ Hi

where a ∈ Ĥ and λ ∈ Z2s [ω].
Therefore,

φs(u) + φs(v) = φs(a) ∈ H.

So, H is linear.

6. Kernel of Z2s [ω]-Linear GH Codes

The method to find the kernel of codes over Z2s is given in Section 4 of [6, 17]. This
section is devoted to establishing several results related to the kernel of Z2s [ω]-linear codes.

Assume A(t1,...,ts) represents the generator matrix of Ĥ(t1,...,ts) and denote wi as the
i-th row vector of A(t1,...,ts). By established construction, w1 = 1 and ord(wi) ≤ ord(wj)
if i > j.

We introduce σ ∈ {1, . . . , s} as the integer satisfying the condition that ord(w2) =
2s+1−σ. Note σ = 1 if t1 > 1, and

σ = min{i : ti > 0, i ∈ {2, . . . , s}} if t1 = 1.

In this case, if σ = s, the code is Ĥ(1,0,...,0,ts), which is linear.
Let u = (u1, . . . , un) ∈ Zn

2s [ω] and

[uj,0, uj,1, . . . , uj,s−1]2

be the 2-ary expansion of uj , where j ∈ {1, . . . , n}.
Assume i is an integer such that i ∈ {1, . . . , s − 1}. Then ui denotes the vector in

which the j-th coordinate corresponds to the i-th element of the 2-ary expansion of uj ,
that is,

ui = (u1,i, . . . , un,i) ∈ Zn
2 [ω].

Proposition 6.1 Let Ĥ = Ĥ(t1,...,ts) be the Z2s [ω]-additive Hadamard code of type
(n; t1, . . . , ts) such that φs(Ĥ) is nonlinear. Define Ĥb as the subcode of Ĥ consisting of
all codewords of order two. Let

B =

{
{2p}σ−2

p=0 if σ ≥ 2,

∅ if σ = 1.

Then,

⟨φs(Ĥb), φs(B), φs

( s−2∑
i=0

2i
)
⟩ ⊆ K(φs(Ĥ))



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 18 of 32

and

ker(φs(Ĥ)) ≥ σ +
s∑

i=1

ti.

Proof: Let H = φs(Ĥ) and r =
∑s

i=1 ti. Let

Q = {ord(wj/2) ·wj}rj=0.

Since Ĥb includes all elements of Ĥ with order two, the set φs(Q) serves as a basis for the
binary linear subcode Hb = φs(Ĥb) of H.

By Corollary 3.6, for all b ∈ Ĥb and u ∈ Ĥ, we have

φs(b) + φs(u) = φs(b + u) ∈ H,

and therefore, Hb ⊆ K(H).
Assume σ ≥ 2. Now, we prove that φs(2

p) ∈ K(H) for all p ∈ {0, . . . , σ − 2}.
Equivalently, we show that

φs(2
p) + φs(u) ∈ H for all u ∈ Ĥ.

If u ∈ Ĥ, then u = µ · 1+ u′, where µ ∈ Z2s [ω] and ord(u′) ≤ ord(w2) = 2s+1−σ.
Let u = (u1, . . . , un) ∈ Zn

2s [ω] and

[ui,0, ui,1, . . . , ui,s−1]2

be the binary expansion of ui, i ∈ {1, . . . , n}.
Let

[µ0, µ1, . . . , µs−1]2

be the binary expansion of µ ∈ Z2s [ω].
Note that if v ∈ Z2s [ω] is of order 2

i, then its binary expansion is of the form

[0, . . . , 0, vs−i, vs−i+1, . . . , vs−1]2.

Since p ∈ {0, . . . , σ − 2} and ord(u′) ≤ 2s+1−σ, we have

u(p) = (u1,p, . . . , un,p) = (µp, . . . , µp).

By Corollary 3.4, we have

φs(2
p) + φs(u) = φs(2

p + u− 2p+1tp),

where

tp =

{
1 if µp ≥ 10,

0 otherwise.

Therefore, 2p+1tp is either 0 or 2p+1. In both cases, 2p+1tp ∈ Ĥ, so

φs(2
p) + φs(u) = φs(2

p + u− 2p+1tp) ∈ H.



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 19 of 32

Next, it can be easily verified that

φs

(
s−2∑
i=0

2i

)
∈ K(H).

Finally, it remains to verify that the elements of the set

{φs(Ĥb), φs(B), φs

(
s−2∑
i=0

2i

)
}

are linearly independent.
Due to the block upper-triangular structure of the generator matrix, it is straightfor-

ward to verify that the codewords in φs(Q) are linearly independent from the codewords
in

{φs(B), φs

(
s−2∑
i=0

2i

)
}.

Note σ ≤ s since H is nonlinear. Thus, by applying Lemma 3.4, we readily conclude
that the codewords in

{φs(B), φs

(
s−2∑
i=0

2i

)
}

are linearly independent, which implies that the dimension of their linear span is σ+ r, so

ker(H) ≥ σ + r.

Lemma 6.1: Let v, µ ∈ Z2s [ω]. Then,

v ⊙2 µ =
s−1∑
i=0

(
v ⊙2 µi2

i
)
,

where
[
µ0, . . . , µs−1

]
2
is the 2-ary expansion of µ.

Proof: Let v ∈ Z2s [ω] and
[
v0, . . . , vs−1

]
2
be its 2-ary expansion. From the definition,

we have

v ⊙2 µ = v ⊙2

s−1∑
i=0

µi2
i =

s−1∑
i=0

ti2
i,

where

ti =

{
1 if vi + µi ≥ 2,

0 otherwise.

Note that
ti2

i = v ⊙2 µi2
i,



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 20 of 32

so

v ⊙2

s−1∑
i=0

µi2
i =

s−1∑
i=0

(
v ⊙2 µi2

i
)
.

Lemma 6.2: Let H = H(t1,...,ts) be the Z2s [ω]-additive Hadamard code of type
(n; t1, . . . , ts). Define

N =

{
s−1∑
i=0

µi2
i : µi ∈ Z2[ω]

}
\

{
s−1∑
i=0

2i

}
if σ ≤ s− 1.

Then,
φs(N ) ∩K

(
φs(H)

)
= {0}.

Proof: Let H = φs(H). Suppose

u =
s−1∑
i=0

µi2
i ∈ N such that φs(u) ∈ K(H).

We want to show that u = 0.
Based on the construction, the second row w2 of A(t1,...,ts) is a 2t−2s+σ-fold replication

of
v = 2σ−1

(
00, . . . , 0, 2s+1−σ − 1, 10, . . . , 2s+1−σ − 1, 2s+1−σ − 1

)
,

and ord(w2) = 2s+1−σ.
By Corollary 3.2, we have

φs(w2) + φs(u) = φs

(
w2 + u− 2(w2 ⊙2 u)

)
.

Since φs(u) ∈ K(H), it follows that 2(w2 ⊙2 u) ∈ H.
Write

w2 = (w1, w2, . . . , wn),

and let
[wj,0, wj,1, . . . , wj,s−1]2

be the 2-ary expansion of wj , for j ∈ {1, . . . , n}.
By Lemma 6.1,

2(w2 ⊙2 u) = 2

s−2∑
i=σ−1

(
w2 ⊙2 µi2

i
)
= 2

s−2∑
i=σ−1

Ri2
i ∈ H,

where
Ri = (r1,i, r2,i, . . . , rn,i),



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 21 of 32

and

rj,i =

{
1 if wj,i + µi ≥ 2,

0 otherwise.

Let τ =
∑s

i=1 ti. Since σ ≤ s− 1, we have τ ≥ 2.
If τ = 2, then H has length 2s+1−σ, and the only vectors in A(t1,...,ts) are 1 and w2 = v.
If τ ≥ 3, for i ∈ {3, . . . , τ}, the i-th row wi of A

(t1,...,ts) has zeros in its first 2s+1−σ

coordinates.
Since σ ≤ s − 1 and τ ≥ 2, every element of H, when restricted to the first 2s+1−σ

coordinates, takes the form
µ1 · 1+ µ2 · v

for some µ1, µ2 ∈ Z2s [ω].
Now,

2
s−2∑

i=σ−1

Ri2
i

restricted to the first m = 2s+1−σ coordinates is

2

s−2∑
i=σ−1

R′
i2

i,

where
R′

i = (t1,i, t2,i, . . . , tm,i).

Therefore, we want µ1, µ2 ∈ Z2s [ω] such that

2
s−2∑

i=σ−1

R′
i2

i = µ1 · 1+ µ2 · v.

Since the initial entry of v is zero, the first coordinate of v(i) is zero for all i ∈
{0, . . . , s− 1}. Thus,

µ1 = 0,

and

2
s−2∑

i=σ−1

R′
i2

i = µ2v.

Note that

v =
s−1∑
i=0

v(i)2i =
s−1∑

i=σ−1

v(i)2i.

Let

a = 2

s−2∑
i=σ−1

R′
i2

i, b = µ2v.



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 22 of 32

Suppose µ2 ∈ A = {0, 2s−σ+1}. Then b = 0. Given the existence of some µi0 ̸= 0, the
vector R′

i0
has at least one nonzero coordinate, so a ̸= 0, a contradiction.

On the other hand, if µ2 ∈ Z2s [ω] \A, let

a(i) = (a1,i, a2,i, . . . , an,i), σ ≤ i ≤ s− 1,

where aj,i ∈ {0, 1} for all j ∈ {1, . . . ,m} and i ∈ {σ, . . . , s− 1}.
Since

v = 2σ−1(00, . . . , 0, 2s+1−σ − 1, 10, . . . , 2s+1−σ − 1, 2s+1−σ − 1),

there exists some i1 ∈ {σ, . . . , s − 1} such that the coordinates of b(i1) do not belong to
{0, 1}, a contradiction.

Therefore, if u ̸= 0, then

2(w2 ⊙2 u) = µ1 · 1+ µ2 · v,

and hence u = 0.
Lemma 6.3: Let Ĥ = Ĥ(t1,...,ts) be the Z2s [ω]-additive GH code of type (n; t1, . . . , ts).

Let wi be the ith row of A
(t1,t2,...,ts)
2 and τ =

∑s
i=1 ti. Define

Ξ =

{
v =

τ−ts∑
i=2

µiwi : µi ∈ Z2s [ω], ord(v) > 2

}
,

N =

{
s−1∑
i=0

µi2
i : µi ∈ Z2[ω] \

{
s−1∑
i=0

2i

}}
if σ ≤ s− 1,

and
Ξ +N = {vΞ + vN : vΞ ∈ Ξ ∪ {0},vN ∈ N}.

Then
φs(Ξ +N ) ∩K(φs(Ĥ)) = {0}.

Proof: Let H = φs(Ĥ), which has length N = 22t = n · 22(s−1). By Lemma 5, we
know that

φs(N ) ∩K(H) = {0},

now we prove that
φs(Ξ) ∩K(H) = ∅.

Let v =
∑τ−ts

i=2 µiwi ∈ Ξ. Since ord(v) > 2 and ord(wi) ≤ 2s+1−σ, we have ord(v) = 2r

for some 2 ≤ r ≤ s+1−σ. Through the iterative construction of A(t1,...,ts), it is clear that
all elements of Z2s [ω] whose order is 2r or less appear as a coordinate of v.

Let
[vj,0, vj,1, . . . , vj,s−1]2



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 23 of 32

be the 2-ary expansion of vj , for j ∈ {1, . . . , n}. By Corollary 4, we have

φs(v) + φs(2
s−r) = φs

(
v + 2s−r − 2s−r+1Rs−r

)
,

where
Ts−r = (t1,(s−r), t2,(s−r), . . . , tn,(s−r)),

and for j ∈ {1, . . . , n},

tj,(s−r) =

{
1, if vj,(s−r) ≥ 1,

0, otherwise.

It suffices to show that
2s−r+1Ts−r /∈ Ĥ

to prove that φs(v) /∈ K(H).
Since

v =

τ−ts∑
i=2

µiwi = (v1, v2, . . . , vn)

and ord(v) = pr for some 2 ≤ r ≤ s + 1 − σ, according to the construction, v contains
each element of 2s−1Z2s [ω] exactly α times, α ≥ 0, and the remaining n− 4α coordinates
come from

Z2s [ω] \ 2s−1Z2s [ω].

So,

wtH
(
φs(2

s−r+1Rs−r)
)
≤ (n− 4α) · 3 · 22(s−2) < 3n · 22(s−2) =

3N

4
= d(H).

Therefore,
φs(v) /∈ K(H),

and
φs(Ξ) ∩K(H) = ∅.

We now proceed to show that

φs(Ξ +N ) ∩K(φs(Ĥ)) = {0}.

Let
v = vΞ + vN ∈ Ξ +N \ {0},

where vΞ ∈ Ξ and vN ∈ N . We previously proved that

φs(v) /∈ K(H)

if vΞ = 0 or vN = 0. Hence, assume vΞ ̸= 0 and 0N ̸= 0.
We know

vN = (v, . . . , v).



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 24 of 32

Let
[v0, v1, . . . , vs−1]2

be the 2-ary expansion of v.
Consider

vN1 and vN2

as the elements of Z2s [ω] having 2-ary expansions

[0, . . . , 0, vs−r, . . . , vs−1]2 and [v0, . . . , vs−r−1, 0, . . . , 0]2,

respectively.
Then,

vN = vN1 + vN2 ,

where
vNi = (vNi , . . . , vNi), i ∈ {1, 2}.

Since ord(vΞ) = 2r with 2 ≤ r ≤ s+ 1− σ, the 2-ary expansion of each coordinate of
vΞ takes the form

[0, . . . , 0, vΞ,(s−r), . . . , vΞ,(s−1)]2.

Note that
ord(vN1) ≤ ord(vΞ)

by construction.
It follows that

2
(
vN2 ⊙2 2

s−r
)
= 0.

Therefore,

wtH
(
φs(2(v ⊙2 2

s−r))
)
= wtH

(
φs(2((vΞ + vN1)⊙2 2

s−r))
)
.

Since ord(vN1) ≤ ord(vΞ), there exists a permutation of coordinates π satisfying

π(vΞ + vN1) = vΞ.

Thus,

wtH
(
φs(2((vΞ + vN1)⊙2 2

s−r))
)
= wtH

(
φs(2(vΞ ⊙2 2

s−r))
)
.

Since ord(vΞ) = pr with 2 ≤ r ≤ s + 1 − σ, as in the previous case, this leads to a
contradiction.

Therefore,
φs(v) /∈ K(H) and φs(Ξ +N ) ∩K(H) = {0}.

Theorem 6.1: Let Ĥ = Ĥ(t1,...,ts) be the Z2s [ω]-additive Hadamard code of type
(n; t1, . . . , ts) such that φs(Ĥ) is nonlinear. Define Ĥb as the subcode of Ĥ consisting of
all the codewords of order two. Let

B =

{
{2p}σ−2

p=0 if σ ≥ 2,

∅ if σ = 1.



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 25 of 32

Then,

⟨φs(Ĥb), φs(B), φs

(
s−2∑
i=0

2i

)
⟩ ⊆ K(φs(Ĥ)),

and

ker(φs(Ĥ)) = σ +

s∑
i=1

ti.

Proof: The conclusion directly follows from Proposition 6.1 and Lemma 6.3.
Corollary 6.1: Let Ĥ = Ĥ(t1,...,ts) be the Z2s [ω]-additive Hadamard code of type

(n; t1, . . . , ts) such that φs(Ĥ) is nonlinear. Let wi be the ith row of A
(t1,t2,...,ts)
2 and

τ =
∑s

i=1 ti. Let

Q = {ord(wj/2)wj}rj=0, B =

{
{2p}σ−2

p=0 if σ ≥ 2,

∅ if σ = 1.

Then {φs(Q), φs(B), φs(
∑s−2

i=0 2
i)} forms a basis for itK(φ(Ĥ)).

Example 6.1: Let it H(2,0,0) be the Z8[ω]-linear Hadamard code discussed in Example
4. According to Theorem 3.4,

ker(H(2,0,0)) = 3.

By Corollary 6.1, K(H(2,0,0)) can be constructed from a basis. To begin with, we have
that

Q = {40, (00, 04, . . . , 00, 04, 40, 44, . . . , 40, 44, 00, 04, . . . , 40, 44)}.

Since σ = 1, in this case B = ∅. Thus,

K(H(2,0,0)) = ⟨φs(40), φs(00, 04, . . . , 00, 04, 40, 44, . . . , 40, 44, 00, 04, . . . , 40, 44), φs(30)⟩.

7. Classification of Z2s [ω]-Linear Hadamard Codes

Our discussion here is aimed at some aspects of classifying Z2s [ω]-linear codes for
length 22t for t ≥ 3 and s > 2, but we realize that dimension of the kernel alone is not
sufficient for a full classification. Theorem 3 states that for any t ≥ 3 and s > 2, there are
at most two Z2s [ω]-linear codes of length 22t, namely; H(1,0,...,0,1,ts) and H(1,0,...,0,ts) which
are linear. Consequently, our focus can be then placed on t ≥ 5 and 2 ≤ s ≤ t− 2 in order
to classify the nonlinear codes.

Theorem 7.1 Let At,s denote the number of inequivalent Z2s [ω]-linear Hadamard
codes of length 22t. Then,

At,s =


0 if t ≥ 3 and s ≥ t+ 2,

1 if t ≥ 3 and s ∈ {t− 1, t, t+ 1},
1 if t = 4 and s = 2,



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 26 of 32

and the Z2s [ω]-linear Hadamard code is linear when At,s = 1. Moreover, if t ≥ 5 and
2 ≤ s ≤ t − 2, then At,s ≥ 2, and there exist one linear code and at least one nonlinear
code.

Proof: If t ≥ 3 and s ≥ t+ 2, then for the equation

t =

(
s∑

i=1

(s− i+ 1)ti

)
− 1,

with t1 ≥ 1, there are no nonnegative integer solutions, so At,s = 0.
If t ≥ 3 and s = t + 1, there is only one solution (t1, . . . , ts) = (1, 0, . . . , 0). If t ≥ 3

and s = t, there is exactly one solution (1, 0, . . . , 0, 1). If t ≥ 3 and s = t − 1, there are
two solutions (1, 0, . . . , 0, 2) and (1, 0, . . . , 0, 1, 0). Notably, when t = 3 and s = 2, both
solutions are (1, 2) and (2, 0). By Theorem 3.3, for all the above solutions, we obtain a
linear code H(t1,...,ts).

Finally, if t ≥ 5 and 2 ≤ s ≤ t−2, the solutions (1, 0, . . . , 0, t−s+1) and (1, 0, . . . , 0, 1, t−
s− 1) always exist, yielding a linear code. In these cases, there is at least one additional
solution. If s = 2,

At,s =

⌊
t− 1

2

⌋
≥ 2 since t ≥ 5.

On the other hand, if s = 3, (2, 0, . . . , 0, t− 2s+ 1) is a solution because t ≥ 2s− 1 when
t ≥ 5; and if s ≥ 4, (1, 0, . . . , 0, 1, 0, t − s − 2) is a solution. Therefore, for all the cases,
At,s ≥ 2 by Theorem 5.2.

Example 7.1: The Z8[ω]-linear Hadamard codes of length 22t = 65536, listed below,
are:

H(1,0,6), H(1,1,4), H(1,2,2), H(1,3,0), H(2,0,3), H(2,1,1), and H(2,1,1).

Both of the first two are equivalent because they are linear codes by Theorem 3. According
to Theorem 4, the other codes have kernel dimensions of 7, 6, 6, 5, and 4. Therefore, from
this invariant, we conclude that all these codes are different except H(1,3,0) and H(2,0,3)

that share identical kernel dimensions.
We have observed in some instances that codes defined over Z2s and codes over

Z2s [ω] have the same rank and kernel dimension, meaning that rank(H(1,3,0)) = 12, and
rank(H(2,0,3)) = 11, which gives their non-equivalence. As a result, in the case of the
Z2s [ω]-linear Hadamard codes of length 22t = 65536, the rank gives a complete classifica-
tion that does not require consideration of the kernel.

Example 7.2: Theorem 6.1 verifies that for all 5 ≤ t ≤ 7 and 2 ≤ s ≤ t − 2, the
nonlinear Z2s [ω]-linear Hadamard codes of length 22t have different kernel dimensions and
we are able to classify these codes according to this invariant. Similar results apply for
t = 8, 9, 10, and 11; exceptions occur for certain values of s in each case. In the particular
cases of t and s, classification according to kernel analysis provides only partial results.

It was found that for codes over Z2s and codes over Z2s [ω], the rank and dimension
of kernel are equal. The software Magma can be used to determine the rank and kernel



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 27 of 32

dimension for any 5 ≤ t ≤ 11 and 2 ≤ s ≤ t− 2 [6, 17]. Tables 2 and 5 give the values of
(t1, . . . , ts) together with the pair (r, k), where r is the rank and k the dimension of the
kernel for all nonlinear Z2s [ω]-linear Hadamard codes of length 22t for 5 ≤ t ≤ 10. Such
tables show that each length 22t code has exactly one rank value when 5 ≤ t ≤ 10 and
2 ≤ s ≤ t − 2 is fixed. Thus, all codes in such situations are different, allowing only the
rank of the code to be classified, from which we obtain the following assertion.

Table 2: Rank and dimension of kernel for all nonlinear Z2s [ω]-linear Hadamard codes of length 22t

t = 5 t = 6 t = 7

(t1, . . . , ts) (r, k) (t1, . . . , ts) (r, k) (t1, . . . , ts) (r, k)

Z4[ω] (3,0) (7,4) (3,1) (8,5) (3,2) (9,6)

(4,0) (11,5)

Z8[ω] (2,0,0) (8,3) (1,2,0) (8,5) (1,2,1) (9,6)

(2,0,1) (9,4) (2,0,2) (10,5)

(2,1,0) (12,4)

Z16[ω] (1,1,0,0) (9,4) (1,0,2,0) (9,6)

(1,1,0,1) (10,5)

(2,0,0,0) (14,3)

Z32[ω] (1,0,1,0,0) (10,5)

Theorem 7.2: Let At,s represent the number of inequivalent Z2s [ω]-linear Hadamard
codes of length 22t. Then, for any t ≥ 3 and 2 ≤ s ≤ t− 1,

At,s ≤

∣∣∣∣∣
{
(t1, . . . , ts) ∈ Ns : t =

(
s∑

i=1

(s− i+ 1)ti

)
− 1, t1 ≥ 1

}∣∣∣∣∣− 1.

Moreover, for any values of t within the range [3, 11] and s in the range [2, t − 1],
including the endpoints, this bound is sharp. Using the outputs of Theorems 5.1 and
5.2, we give the following Table 3 that lists the number of nonequivalent Z2s [ω]-linear
Hadamard codes of length 22t where 3 ≤ t ≤ 11 and 2 ≤ s ≤ 9. Classification is inadequate
only when considering the kernel dimension in the highlighted cases in bold. However, in
all cases, the aforesaid rank turns out to be a good method of classification.

There appear to be Z4[ω]-linear Hadamard codes that do not exist as equivalent Z2s [ω]-
linear Hadamard codes, for s > 2.

Example 7.3: Let us consider H(2,0,0) as the Z8[ω]-linear Hadamard code of length
1024. From Theorem 3, we know that ker(H(2,0,0)) = 3, i.e., H(2,0,0) cannot be linear.

There are acknowledged to be three Z4[ω]-linear Hadamard codes of length 1024 given
by H(1,4), H(2,2), and H(3,0). The first two codes have a linear structure, while the last one
is nonlinear, and by Theorem 5.2 it is shown that ker(H(3,0)) = 4. Consequently, there
does not exist a Z4[ω]-linear Hadamard code equivalent to the Z8[ω]-linear Hadamard code
H(2,0,0).

Example 7.4: It is apparent from Table 2 that for t = 5, we have only two nonlinear
Z2s [ω]-linear Hadamard codes, namely H(3,0) and H(2,0,0). Based on Example 7.3, the



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 28 of 32

Table 3: Number At,s of nonequivalent Z2s [ω]-linear Hadamard codes of length 22t

t 3 4 5 6 7 8 9 10 11

Z4[ω] 1 1 2 2 3 3 4 4 5
Z8[ω] 1 1 2 3 4 6 7 9 11
Z16[ω] 1 1 1 2 4 5 8 10 14
Z32[ω] 0 1 1 1 2 4 6 9 12
Z64[ω] 0 0 1 1 1 2 4 6 10
Z128[ω] 0 0 0 1 1 1 2 4 6
Z256[ω] 0 0 0 0 1 1 1 2 4
Z512[ω] 0 0 0 0 0 1 1 1 2

codes are shown to be distinct because they differ in the kernel’s dimension.
More examples can be seen when t is an odd number. For example, based on Tables

2 and 5, when t = 7, t = 9, and t = 11, we observe that the Z4[ω]-linear Hadamard codes
H(4,0), H(5,0), and H(6,0) do not coincide in equivalence with any Z2s [ω]-linear Hadamard
codes of the same length and s > 2, under both the rank and the dimension of the kernel.

It has been established that for Z2-linear Hadamard codes, the lower bounds K (the
kernel dimension) and RK (the kernel dimension and the rank) are known. In this paper,
we have shown that for both Z2s- and Z2s [ω]-linear Hadamard codes, the upper bounds
K and RK are the same. The Table 4 contains bounds for the range 3 ≤ t ≤ 11.

Table 4: Bounds for the number At of nonequivalent Z2s [ω]-linear Hadamard codes of length 22t

t 3 4 5 6 7 8 9 10 11

Lower bound K 1 1 3 3 5 5 7 7 9
Lower bound RK 1 1 3 3 6 7 11 13 20
Upper bound 1 1 3 5 10 16 26 38 57

By studying all nonequivalent Z2s [ω]-linear Hadamard codes of length 22t, it is possible
to produce an easy upper bound calculation if t and s are given. Table 4 shows the values
for all 3 ≤ t ≤ 11.

Theorem 7.3: Let A(t,s) be defined as the number of nonequivalent Z2s [ω]-linear
Hadamard codes of length 22t. Let At denote the total number of distinct Z2s [ω]-linear
Hadamard codes of length 22t for s ≥ 2. Then,

At ≤
t−2∑
s=2

(
A(t,s) − 1

)
+ 1.

Theorem 7.4: For lengths 22t, where t = 3, 4, 5, 6 and 7, there exist exactly 1, 1, 3,
3, and 6 distinct nonequivalent Z2s [ω]-linear Hadamard codes, respectively.

8. Conclusion and Future Directions

Our research created and studied Generalized Hadamard (GH) codes using Eisenstein
local rings Z2s [ω] to develop their complete set of algebraic and combinatorial properties.



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 29 of 32

Our use of Eisenstein integers established a Gray map to switch from these codes to binary
numbers for expert study. Our research revealed specific rules to distinguish Z2s [ω]-linear
GH codes that are linear from those that are not. Our research into the kernel properties
helps reveal how these codes differ from one another. The theory proves that Eisenstein
fields offer effective ways to build useful and space-saving error correction codes.

The future of research should study different families of error-correcting codes using
Eisenstein algebraic rings and their relatives, such as cyclic, quasi-cyclic, and consta-cyclic
codes. Studying automorphism groups and decoding methods for GH codes with Z2s [ω]
as field structure will lead to theoretical advancement and practical applications.

Acknowledgements

This Research is supported by Universidad Pedagógica y Tecnológica de Colombia 578
(SGI 3725) and Minciencias (Conv. 934).

Tribute

We would like to express our heartfelt gratitude to our beloved supervisor, Professor Dr.
Tariq Shah (late), whose exceptional guidance, profound expertise, and steadfast support
were instrumental in shaping our academic journey. His mentorship not only nurtured our
growth as researchers in algebra, number theory, coding theory, and cryptography but also
profoundly influenced our personal and professional development. His legacy of wisdom,
integrity, and inspiration continues to guide us. May his soul rest in eternal peace.

Figure 1: Prof. Dr. Tariq Shah



Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 30 of 32

Data availability

All the data is given in this study.

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Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6409 32 of 32

Table 5: Rank and kernel for all nonlinear Z2s [ω]-linear Hadamard codes of length 2t

t = 8 t = 9 t = 10

(t1, . . . , ts) (r, k) (t1, . . . , ts) (r, k) (t1, . . . , ts) (r, k)

Z4[ω] (3,3) (10,7) (3,4) (11,8) (3,5) (12,9)

(4, 1) (12, 6) (4, 2) (13, 7) (4,3) (14,8)

(5, 0) (16, 6) (5,1) (17,7)

Z8[ω] (1,2,2) (10,7) (1,2,3) (11,8) (1,2,4) (12,9)

(1, 3, 0) (12, 6) (1,3,1) (13,7) (1,3,2) (14,8)

(2, 0, 3) (11, 6) (2, 0, 4) (12, 7) (1, 4, 0) (17,7)

(2,1,1) (13,5) (2,1,2) (14,6) (2,0,5) (13,8)

(3,0,0) (17,4) (2,2,0) (17,5) (2,1,3) (15,7)

(3,0,1) (18,5) (2,2,1) (18,6)

(3,0,2) (19,6)

(3,1,0) (24,5)

Z16[ω] (1, 0, 2, 1) (10, 7) (1,0,2,2) (11,8) (1,0,2,3) (12,9)

(1, 1, 0, 2) (11, 6) (1,0,3,0) (17,7) (1,0,3,1) (14,8)

(1, 1, 1, 0) (13, 5) (1,2,0,0) (18,5) (1,1,0,4) (13,8)

(2, 0, 0, 1) (15, 4) (1,1,0,3) (12,7) (1,1,1,2) (15,7)

(1, 1, 1, 1) (14,6) (1,1,2,0) (18,6)

(2, 0, 0, 2) (16,5) (1,2,0,1) (19,6)

(2,0,1,0) (20,4) (2,0,0,3) (17,6)

(2, 0, 1, 1) (21,5)

(2,1,0,0) (28,4)

Z32[ω] (1, 0, 0, 2, 0) (10, 7) (1, 0, 0, 2, 1) (11, 8) (1, 0, 0, 2, 2) (12, 9)

(1, 0, 1, 0, 1) (11, 6) (1, 0, 1, 0, 2) (12, 7) (1, 0, 0, 3, 0) (14,8)

(1, 1, 0, 0, 0) (15, 4) (1, 0, 1, 1, 0) (14, 6) (1, 0, 1, 0, 3) (13,8)

(1, 1, 0, 0, 1) (16, 5) (1, 0, 1, 1, 1) (15,7)

(2, 0, 0, 0, 0) (26, 3) (1, 0, 2, 0, 0) (19,6)

(1, 1, 0, 0, 2) (17,6)

(1, 1, 0, 1, 0) (21, 5)

(2, 0, 0, 0, 1) (27, 4)

Z64[ω] (1, 0, 0, 1, 0, 0) (11, 6) (1, 0, 0, 0, 2, 0) (11, 8) (1, 0, 0, 0, 2, 1) (12, 9)

(1, 0, 0, 1, 0, 1) (12, 7) (1, 0, 0, 1, 0, 2) (13, 8)

(1, 0, 1, 0, 0, 0) (16, 5) (1, 0, 0, 1, 1, 0) (15, 7)

(1, 0, 1, 0, 0, 1) (17, 6)

(1, 1, 0, 0, 0, 0) (27, 4)

Z128[ω] (1, 0, 0, 0, 1, 0, 0) (12, 7) (1, 0, 0, 0, 0, 2, 0) (19, 2)

(1, 0, 0, 0, 1, 0, 1) (13, 8)

(1, 0, 0, 1, 0, 0, 0) (17, 6)

Z256[ω] (1, 0, 0, 0, 0, 1, 0, 0) (13, 8)