JACODESMATH / ISSN 2148-838X

J. Algebra Comb. Discrete Appl.
2(2) • 75-84

Received: 29 October 2014; Accepted: 2 February 2015
DOI 10.13069/jacodesmath.17537

Journal of Algebra Combinatorics Discrete Structures and Applications

On a class of repeated-root monomial-like abelian codes

Research Article

Edgar Martínez-Moro1∗, Hakan Özadam2,3∗∗, Ferruh Özbudak2§, Steve Szabo3∗∗∗

1. Institute of Mathematics and Applied Mathematics Department University of Valladolid, Castilla, Spain

2. Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University İnönü
Bulvarı, 06531, Ankara, Turkey

3. University of Massachusetts, Medical School Worcester, Massachusetts

4. Department of Mathematics and Statistics, Eastern Kentucky University Richmond, KY, USA

Abstract: In this paper we study polycyclic codes of length ps1 × · · · × psn over Fpa generated by a single
monomial. These codes form a special class of abelian codes. We show that these codes arise from
the product of certain single variable codes and we determine their minimum Hamming distance.
Finally we extend the results of Massey et. al. in [10] on the weight retaining property of monomials
in one variable to the weight retaining property of monomials in several variables.

2010 MSC: 94B15,11T71,11T55

Keywords: Repeated-root Cyclic code, Abelian code, Weight-retaining property

1. Introduction

Cyclic codes are said to be repeated-root when the codeword length and the characteristic of the
alphabet are not coprime. Despite that it has been proved that in general they are asymptotically bad
in some cases repeated-root cyclic codes are optimal and they have interesting properties. Massey et. al.
have shown in [10] that cyclic codes of length p over a finite field of characteristic p are optimal. There
also exist infinite families of repeated-root cyclic codes in even characteristic according to the results of
[14]. Also in [10] it has been pointed out that some repeated-root cyclic codes can be decoded using a
very simple circuitry. Among other studies on repeated-root cyclic codes with several different settings
are [1, 2, 7, 8, 11, 12, 14].

∗ E-mail: edgar@maf.uva.es
∗∗ E-mail: ozhakan@metu.edu.tr
§ E-mail: ozbudak@metu.edu.tr
∗∗∗ E-mail: steve.szabo@eku.edu

75



Monomial-like codes

Contrary to the simple-root case, there are repeated root cyclic codes of the form
(
f(x)i

)
where

i > 1. Specifically, all cyclic codes of length ps over a finite field of characteristic p are generated by a
single “monomial” of the form (x− 1)i, where 0 ≤ i ≤ ps (see [2, 11]). In this paper, as a generalisation
of these codes to several variables, we study cyclic codes of the form

I(i1,...in) =
(
(x1 − 1)i1 · · ·(xn − 1)in

)
⊂R =

Fpa [x1, . . . ,xn](
x
ps1
1 − 1, . . . ,x

psn
n − 1

), (1)
i.e. I(i1,...in) is the ideal of R generated by a single monomial of the form (x1 − 1)

i1 · · ·(xn − 1)in.
This paper is organised as follows. First we introduce some notation, give some definitions and prove

some structural properties of the ambient space of a particular class of abelian codes in Section 2. In
Section 3, we show thatmonomial like codes arise from product codes and we determine their Hamming
distance. We describe their duals which yields a parity check matrix for these codes. In Section 4, we
explain the relationship of the Hasse derivative with the dual of this type of codes. Finally in Section 5,
we generalise the weight retaining property of monomials in single variable to the multivariable case.

2. The Ambient Space

Throughout the paper, we consider the finite ring

R =
Fpa [x1, . . . ,xn](

x
ps1
1 − 1, . . . ,x

psn
n − 1

) (2)
as the ambient space of the codes to be studied unless otherwise stated. It is a well known fact that R
is a local ring with maximal ideal (x1 − 1, . . . ,xn − 1). We define

L = {(α1,α2, . . . ,αn) | 0 ≤ αj < psj, αj ∈ Z for all 1 ≤ j ≤ n}. (3)

The elements of R can be identified uniquely with the polynomials of the form

f(x1, . . . ,xn) =
∑

(α1,α2,...,αn)∈L

f(α1,α2,...,αn)x
α1
1 x

α2
2 · · ·x

αn
n , (4)

so throughout the paper, we identify the equivalence class

f(x1, . . . ,xn) +
(
x
ps1

1 − 1,x
ps2

2 − 1, . . . ,x
psn
n − 1

)
with the polynomial f(x1, . . . ,xn). We shall consider a repeated-root code as just an ideal C of R. The
length of the code is ps1 × ps2 × ···× psn and the support of a codeword f(x1, . . . ,xn) ∈ C is the set
supp(f) = {(α1,α2, . . . ,αn) ∈ L | f(α1,α2,...,αn) 6= 0}. The Hamming weight of f(x1, . . . ,xn) is defined
as w(f(x1, . . . ,xn)) = |supp(f)|, i.e. the number of nonzero coefficients of f(x1, . . . ,xn). The minimum
Hamming distance of the code C is defined as

d(C) = min{w(f(x1, . . . ,xn)) | f(x1, . . . ,xn) ∈C \{0}}.

3. Monomial-like codes

In this paper we shall study a particular class of the codes over R called monomial-like codes given
by an ideal generated by a single monomial of the form

C(i1,...,in) =
(
(x1 − 1)i1 · (x2 − 1)i2 · · ·(xn − 1)in

)
⊂R. (5)

76



E. Martínez-Moro et al.

Note that not all the ideals in R can be generated by a single monomial of this form.
In one variable case, the minimum Hamming distance of C was computed in [11] and [2]. It turns

out that, in multivariate case, C(i1,...,in) can be considered as a product code of single variable codes.
This decomposition allows us to express the minimum Hamming distance of C(i1,...,in) in terms of the
Hamming distances of cyclic codes of length psj .

Definition 3.1. The product of two linear codes C,C′ over Fpa is the linear code C⊗C′ whose codewords
are all the two dimensional arrays for which each row is a codeword in C and each column is a codeword
in C′.

The following are some well-known facts about the product codes.

1. If C and C′ are [n,k,d] and [n′,k′,d′] codes respectively, then C ⊗C′ is a [nn′,kk′,dd′] code.

2. If G and G′ are generator matrices of C and C′ respectively, then G⊗G′ is a generator matrix of
C ⊗C′, where ⊗ denotes the Kronecker product of matrices and the codewords of C ⊗C′ are seen
as concatenations of the rows in arrays in C ⊗C′.

Theorem 3.2. Let n1,n2 be positive integers and let

R̂ =
Fpa [x,y]

(xn1 − 1,yn2 − 1)
,Rx =

Fpa [x]
(xn1 − 1)

, Ry =
Fpa [y]

(yn2 − 1)
.

Suppose that (x− 1)k1|xn1 − 1 and (y − 1)k2|yn2 − 1. The code

C =
(
(x− 1)k1 · (y − 1)k2

)
⊂R̂

is the product of the codes Cx =
(
(x− 1)k1

)
⊂Rx and Cy =

(
(y − 1)k2

)
⊂Ry, i.e., C = Cx ⊗Cy.

Proof. Let

g(x) = (x− 1)k1 = gk1x
k1 + · · · + g1x + g0, h(y) = (y − 1)k2 = hk2y

k2 + · · · + h1y + h0.

Then

Gx =




0 . . . 0 0 gk1 . . . g1 g0
0 . . . 0 gk1 . . . g1 g0 0
...

...
gk1 . . . g1 g0 0 . . . 0 0


 ,Gy =




0 . . . 0 0 hk2 . . . h1 h0
0 . . . 0 hk2 . . . h1 h0 0
...

...
hk2 . . . h1 h0 0 . . . 0 0




are two generator matrices for Cx and Cy, respectively.

We identify the polynomial f(x,y) =
∑

0≤i<n1,0≤j<n2 cijx
iyj ∈ Fpa [x,y], with the codeword

(cn1−1,n2−2, . . . ,cn1−1,1,cn1−1,0, . . . ,c1,n2−1, . . . ,c1,1,c1,0,c0,n2−1, . . . ,c0,1,c0,0).

The elements of C =
(
(x− 1)k1 (y − 1)k2

)
⊂R̂ are exactly all the Fpa-linear combinations of the elements

of the set

β = {xiyj(x− 1)k1 (y − 1)k2 : 0 ≤ i < n−k1, 0 ≤ j < n−k2}

Now we consider G = Gx ⊗Gy. Using the above identification for the rows of G, we obtain a basis for
Cx ⊗Cy which is equal to β. Thus C = Cx ⊗Cy.

77



Monomial-like codes

Corollary 3.3. Let r1, . . . ,rn, i1, . . . , in be positive integers and let

R′ =
Fpa [x1, . . . ,xn]

(xr11 − 1, . . . ,x
rn
n − 1)

, Rxj =
Fpa [xj](
x
rj
j − 1

).
Suppose that (xj − 1)ij|x

rj
j − 1 for all 1 ≤ j ≤ n. The code

C(i1,...,in) =
(
(x1 − 1)i1 · · ·(xr − 1)ir

)
is the product of the codes Cxj =

(
(xj − 1)ij

)
⊂Rxj , i.e.,

C(i1,...,in) = (· · ·((Cx1 ⊗Cx2 ) ⊗Cx3 ) ⊗···) ⊗Cxn =
n⊗
i=1

Cxi. (6)

Remark 3.4.

1. Note that the tensor product is associative in the sense that there is a natural isomorphism (C ⊗
C′) ⊗C′′ ∼= C ⊗ (C′ ⊗C′′). Thus we can remove all the parenthesis in Equation 6.

2. The reader can identify in Theorem and Corollary 3.3 as a polynomial version of the the fact that for
G a finite p-group such that G = G1×G2×···×Gn and K a field then KG ∼= KG1⊗KG2⊗···⊗KGn
where g = g1g2 . . .gn is mapped to g1 ⊗g2 ⊗···⊗gn.

The previous construction give us a straightforward result for the minimum distance of our codes as
follows.

Corollary 3.5. Let C(i1,...,in) ⊂R then

d(C(i1,...,in)) =
n∏
j=1

d(C(ij )), (7)

where d(C(ij )) is the minimum distance of the code (x
ij
i − 1) in Fpa [x]/(x

ij
i − 1).

Note that d(C(ij )) is explicitly given in [2, Theorem 6.4] and [11, Theorem 1] in terms of p,a and ij.

3.1. Weight hierarchy of some two-variable cases

In some very special two-variable cases we can go slightly further and compute explicitly the whole
weight hierarchy of the code. The r-th generalised Hamming weight dr(C) , 1 ≤ r ≤ k, of a Fp-linear
code C of dimension k is defined as the minimum of the cardinalities of the supports of all the subcodes
(linear subspaces) of dimension r of C. We will define d0(C) = 0. The sequence {dr(C)}kr=0 is called the
Hamming weight hierarchy of C.

Let R′ = Fpa [x]
(xp−1) and C(i1) =

(
(x− 1)i1

)
⊂R′. It was shown in [10, Theorem 5] that C(i1) is a Max-

imum Distance Separable (MDS) code.The weight hierarchy of a MDS code C is completely determined
by its length n and dimension k as dr(C) = n − k + r for r = 1, 2, . . . ,k, see for example [6, Theorem
7.10.7].

Consider now C(i2) =
(
(x2 − 1)i2

)
⊂ Fpa [x2]

(xp2−1)
and let k1,k2 the dimension as Fpa-linear spaces of the

78



E. Martínez-Moro et al.

codes C(i1),C(i2) respectively. Using [13, Theorem 1] and since C(i1) ⊗C(i2) = C(i1,i2) we get

dr(C(i1,i2))

= min{
s∑
i=1

(di(C(i1)) −di−1(C(i1)))dti (C(i2)), 1 ≤ ts ≤ ···≤ t1 ≤ k2,s ≤ k1,
s∑
i=1

ti = r}

= min{d1(C(i1))(i2 + t1) +
s∑
i=2

(i2 + ti), 1 ≤ ts ≤ ···≤ t1 ≤ k2,s ≤ k1,
s∑
i=1

ti = r}

= min{(d1(C(i1)) − 1)(i2 + t1) + r + si2, 1 ≤ ts ≤ ···≤ t1 ≤ k2,s ≤ k1,
s∑
i=1

ti = r}.

Since d1(C1) = i1 + 1 and the minimum value of t1, subject to 1 ≤ ts ≤ ··· ≤ t1 ≤ k2 and
∑s
i=1 ti = r,

is
⌈
r
s

⌉
, we obtain

dr(C(i1,i2)) = min{i1(i2 +
⌈r
s

⌉
) + r + si2, s ≤ k1}, r = 1, 2, . . . ,k1 ·k2. (8)

3.2. Dual codes

Note that the elements of the form
∏n
k=1(xk − 1)

jk with j ∈ Nn form a basis of Fpa [x1, . . . ,xn] and
the elements of this form with jk ≥ psk for some k form a basis of ({xp

sk − 1}nk=1). Let us consider

0 6= f(x1, . . . ,xn) =
∑
j∈L

cj

n∏
k=1

(xk − 1)jk.

Therefore (x1 − 1)i1 · · ·(xn − 1)inf(x1, . . . ,xn) = 0 in R if and only if for every j ∈ L with cj 6= 0 we
have psk ≤ jk + ik for some k if and only if f(x1, . . . ,xn) ∈ ({(x− 1)p

sk−ik}nk=1). This proves that the
annihilator of C(i1,...,in) is ({(x − 1)

psk−ik}nk=1) and the dual of an ideal of R is exactly its annihilator.
Therefore we have proved the following statement.

Theorem 3.6.

C⊥(i1,...,in) = ({(x− 1)
psk−ik}nk=1) ⊂R.

Remark 3.7. Note that the above fact does not hold for arbitrary ideals of algebras of type

F[x1, . . . ,xn]/({xnii − 1}
n
i=1)

and it relies on the fact that the ni = psi.

Let us construct an Fpa-basis for C⊥. This will provide us a generator matrix for C⊥ and hence a
parity check matrix for C.

Let Tk = {(a1, . . . ,an) ∈ Nn | psj − ij ≤ aj < psj if j = k, 0 ≤ aj < psj if j 6= k} and T =
T1 ∪·· ·∪Tn. Let s = s1 + s2 + · · · + sn, it is clear that for e1, . . . ,er pairwise distinct

|Te1 ∩·· ·∩Ter| = ie1 · · ·ier
ps

pse1 · · ·pser
.

Now applying the inclusion-exclusion principle we obtain

|T | =
n∑
j=1

ij
ps

psj
−
∑
j<k

ijik
ps

psjpsk
−··· + (−1)n+1i1 · · ·in

= ps − (ps1 − i1) · · ·(psn − in).

79



Monomial-like codes

Let B = {(x1 − 1)a1 · · ·(xn − 1)an | (a1, . . . ,an) ∈ T}. Clearly the elements of B are Fpa–linearly
independent and |B| = |T |. On the other hand, we know, from Theorem 3.2, that dim(C(i1,...,in)) =
(ps1 −i1) · · ·(psn −in). This implies that dim(C⊥(i1,...,in)) = p

s−dim(C(i1,...,in)) which agree the cardinality
of B, thus the set B is an Fq-basis for C⊥. I.e., if we consider the vector representations of the elements
of B, we obtain a generator matrix for C⊥ and a parity check matrix for C.

4. Duality and the Hasse derivative

In this subsection we will show the natural relation between the Hasse derivative and the dual
of monomial-like of codes. We begin by recalling the Hasse derivative which is used in the repeated-
root factor test. For a detailed treatment of the Hasse derivative, we refer to [4, Chapter 1] and [5,
Chapter 5]. Note that the standard derivative for polynomials over a field of positive characteristic, say
p, is inappropriate because from the pth derivative on, the result is always zero. For this reason, it is
more convenient to work with the Hasse derivative. Sometimes the Hasse derivative is also called the
hyper derivative. Throughout this section, we will use the convention that

(
a
b

)
= 0 whenever b > a.

Let g(x1, . . . ,xn) =
∑
gα1,...,αnx

α1
1 · · ·x

αn
n be a polynomial in Fq[x1, . . . ,xn]. The Hasse derivative of

g(x1, . . . ,xn) in the direction a = (a1, . . . ,an) is defined as

D[a](g(x1, . . . ,xn)) =
∑

gα1,...,αn

(
α1
a1

)
· · ·
(
αn
an

)
xα1−a11 · · ·x

αn−an
n . (9)

We denote the evaluation of D[a](g(x1, . . . ,xn)) at the point (λ1, . . . ,λn) ∈ Fnpa by D[a](g)(λ1, . . . ,λn).
We can express g(x1, . . . ,xn) as

g(x1, . . . ,xn) =
∑

(j1,...,jn)∈S

cj1,...,jn (x1 − 1)
j1 · · ·(xn − 1)jn

where S is a finite nonempty subset of Nn. Let S = U` tP` where

U` = {(j1, . . . ,jn) ∈ S | j` ≥ i`}, P` = {(j1, . . . ,jn) ∈ S | j` < i`}.

Therefore

g(x1, . . . ,xn) =
∑

(j1,...,jn)∈U`

cj1,...,jn (x1 − 1)
j1 · · ·(xn − 1)jn

+
∑

(j1,...,jn)∈P`

cj1,...,jn (x1 − 1)
j1 · · ·(xn − 1)jn,

and the term (x` − 1)i` divides g(x1, . . . ,xn) if and only if cj1,...,jn = 0 for all (j1, . . . ,jn) ∈ P`. Now
suppose that (x` − 1)i` - g(x1, . . . ,xn). Then there is a (æ̂1, . . . , æ̂n) ∈ P` such that cæ̂1,...,æ̂n 6= 0. Hence

D[æ̂](g)(1, . . . , 1) = cæ̂1,...,æ̂n

(
æ̂1
æ̂1

)
· · ·
(

æ̂n
æ̂n

)
6= 0.

Conversely, if (x` − 1)i` divides g(x1, . . . ,xn), then

g(x1, . . . ,xn) =
∑

(j1,...,jn)∈U`

cj1,...,jn (x1 − 1)
j1 · · ·(xn − 1)jn.

Therefore D[~a](g)(1, . . . , 1) = 0 for all ~a = (a1, . . . ,an) with 0 ≤ a` < i`. This proves the following result.

Lemma 4.1. Let g(x1, . . . ,xn) ∈ Fpa [x1, . . . ,xn] and let A` = {~a = (a1, . . . ,an) ∈ Nn | 0 ≤ a` < i`}.
Then (x` − 1)i` divides g(x1, . . . ,xn) if and only if D[~a](g)(1, . . . , 1) = 0 for all ~a ∈ A`.

80



E. Martínez-Moro et al.

As an immediate consequence, we have the following theorem.

Theorem 4.2. Let A` = {~a = (a1, . . . ,an) ∈ Nn | 0 ≤ a` < i`} and A = ∪n`=1A`. Let g(x1, . . . ,xn) ∈
Fpa [x1, . . . ,xn]. We have (x1 −1)i1 · · ·(xn−1)in divides g(x1, . . . ,xn) if and only if D[~a](g)(1, . . . , 1) = 0
for all ~a ∈ A.

Let R be as in (2) and let our code be C(i1,...in) ⊂ R. We know that the polynomial g(x1, . . . ,xn)
is in the code C(i1,...in) if and only if (x1 − 1)

i1 · · ·(xn − 1)in divides g(x1, . . . ,xn). Note that
D[a1,...,an](g)(1, . . . , 1) = 0 if a` ≥ ps` for some 1 ≤ ` ≤ n. Together with this fact, Theorem 4.2
implies the following result.

Theorem 4.3. Let C(i1,...in) ⊂R, and let us define

Q =

n⋃
`=1

{~a = (a1, . . . ,an) ∈ Nn | 0 ≤ a` < i`, 0 ≤ aj < psj for j 6= `}.

Then g(x1, . . . ,xn) ∈C(i1,...in) if and only if D
[~a](g)(1, . . . , 1) = 0 for all ~a ∈ Q.

Now let us fix a monomial order so that x1 > · · · > xn. Let ~a = (a1, . . . ,an) ∈ Q. Consider the
vector

wa =

((
ps1 − 1
a1

)
· · ·
(
psn − 1
an

)
,

(
ps1 − 1
a1

)
· · ·
(
psn−1 − 1
an−1

)(
psn − 1
an

)
, · · ·

(
0

a1

)
, · · ·

(
0

an

))
.

For g(x1, . . . ,xn) ∈R, let ug be the vector representation of the polynomial with respect to the fixed or-
dering. Then the dot product of wa and ug gives us the evaluation of the Hasse derivative of g(x1, . . . ,xn)
at (1, . . . , 1) in the direction ~a, i.e., wa ·ug = D[~a](g)(1, . . . , 1). If we construct the matrix H whose rows
are the vectors wa where ~a ∈ Q and Q is as in Theorem 4.3 then H is an alternative parity check matrix
for the code C(i1,...in) by Theorem 4.3.

5. A generalisation of the weight retaining property

In [10], the so-called weight retaining property of polynomials over finite fields was stated and proved.
This property turned out to be very useful for determining the Hamming distance of cyclic codes.

In this section, we give a generalisation of the weight retaining property to multivariate polynomials.
We prove that the Hamming weight of any Fpa-linear combination of the monomials (x1 −c1)i1 · · ·(xn −
cn)

in is greater than or equal to the Hamming weight of the “minimal” nonzero term, where a “minimal”
term is the one that is not divisible by the rest of the nonzero terms of the summation.

First, we consider the case in one variable which was studied in [10]. The weight retaining property
of (x− c)i is given in the following two theorems.

Theorem 5.1. [10, Theorem 1.1 and Theorem 6.1] Let L be any nonempty finite subset of non-negative
integers with least integer imin and let

f(x) =
∑
i∈L

bi(x− c)i

where c and each bi are nonzero elements of Fpa. Then

w(f(x)) ≥ w((x− c)imin ).

It is not hard to see that Theorem 5.1 is equivalent to the following theorem.

81



Monomial-like codes

Theorem 5.2. [10, Theorem 6.2] For any polynomial Q(x) over Fpa and c ∈ Fpa \{0}, and any non-
negative integer N,

w(Q(x)(x− c)N ) ≥ w((x− c)N )w(Q(c)).

The Hamming weight of the monomial (x− c)i, which is used above, was also determined in [10].
Theorem 5.3. [10, Lemma 1] Let c ∈ Fpa \{0} and let i be an integer with the p-adic expansion

i = ι0 + ι1p + · · ·ιm−1pm−1

where 0 ≤ ι` ≤ p− 1 for all 0 ≤ ` ≤ m− 1. Then

w((x− c)i) = P(i) =
m−1∏
j=0

(ιj + 1).

The following theorem is a generalisation of the Massey’s weight retaining property to n variables.
Its proof is very similar to the proof of [3, Proposition 1.2].

Theorem 5.4. Let ψ ⊂ Nn be a finite set and let (N1,N2, . . . ,Nn) ∈ ψ. Let

f(x1, . . . ,xn) =
∑
β∈ψ

cβ(x1 − c1)β1 (x2 − c2)β2 · · ·(xn − cn)βn ∈ Fpa [x1, . . . ,xn],

where cβ ∈ Fpa \{0}, β = (β1, . . . ,βn) and (x1 −c1)N1 (x2 −c2)N2 · · ·(xn−cn)Nn divides (x1 −c1)β1 (x2 −
c2)

β2 · · ·(xn − cn)βn for every β ∈ ψ. Then

w(f(x1, . . . ,xn)) ≥
n∏
i=1

P(Ni).

Proof. The proof is via induction on n. For n = 1, the claim follows by Theorem 5.1. Now assume
that the claim holds true for n− 1. We can express f(x1, . . . ,xn) as

(xn − cn)Nn (
∑
β∈ψ

c
(0)
β (x1 − c1)

β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1

+(xn − cn)
∑
β∈ψ

c
(1)
β (x1 − c1)

β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1

...
+(xn − cn)r

∑
β∈ψ

c
(r)
β (x1 − c1)

β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1 )

for some non-negative integer r and c(`)β ∈ Fpa. By the induction step, we have

w(
∑
β∈ψ

c
(0)
β (x1 − c1)

β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1 ) ≥ P(N1) · · ·P(Nn−1).

If we express each summand
∑
β∈ψ c

(u)
β (x1 − c1)

β1 (x2 − c2)β2 · · ·(xn−1 − cn−1)βn−1 in the form∑
β∈ψ′ e

(u)
β x

β1
1 x

β2
2 · · ·x

βn−1
n−1 , we get

(xn − cn)Nn (
∑
β∈ψ′

e
(0)
β x

β1
1 x

β2
2 · · ·x

βn−1
n−1 + (xn − cn)

∑
β∈ψ′

e
(1)
β x

β1
1 x

β2
2 · · ·x

βn−1
n−1

. . . + (xn − cn)r
∑
β∈ψ′

e
(r)
β x

β1
1 x

β2
2 · · ·x

βn−1
n−1 ).

82



E. Martínez-Moro et al.

Note that we have just shown that there are at least P(N1) · · ·P(Nn−1) many nonzero e
(0)
β ’s. We

define

hβ(xn) = e
(0)
β + e

(1)
β (xn − cn) + · · · + e

(r)
β (xn − cn)

r.

So

f(x1, . . . ,xn) = (xn − cn)Nn (
∑
β∈ψ

hβ(xn)x
β1
1 · · ·x

βn−1
n−1 ).

There are at least P(N1) · · ·P(Nn−1) many β’s such that hβ(xn) 6= 0. For every such β = (β1, . . . ,βn),
we have

w((xn − cn)Nnhβ(xn)x
β1
1 · · ·x

βn−1
n−1 ) ≥ P(Nn)

because w((xn − cn)Nnhβ(xn)) ≥ P(Nn) as the claim holds for one variable. Hence w(f(x1, . . . ,xn)) ≥
P(N1) · · ·P(Nn−1)P(Nn).

Remark 5.5. This result only applies for polynomials f of a special kind, namely those for which the
set denoted ψ contains (N1, . . . ,Nn). For example, ψ = {(1, 2), (2, 1)} does not have that property. Note
that the condition (N1,N2) ∈ ψ is necessary, consider the following example

f(x1,x2) = (x1 + 1)
4(x2 + 1)

3 + (x1 + 1)
3(x2 + 1)

4

with coefficients in the field of 2 elements. It is easy to check that w(f(x1,x2)) = 14 but P(3) = 4 where
P is the polynomial of Theorem 5.3.

Using Theorem 5.4, we generalise Theorem 5.3 to n variables.

Corollary 5.6. Let Q(x1, . . . ,xn) ∈ Fpa [x1, . . . ,xn], c1, . . . ,cn ∈ Fpa and N1, . . . ,Nn ∈ N. We have

w[Q(x1, . . . ,xn)(x1 − c1)N1 · · ·(xn − cn)Nn ]
≥ w[(x1 − c1)N1 · · ·(xn − cn)Nn ][Q(c1, . . . ,cn)]
= P(N1) · · ·P(Nn)wH[Q(c1, . . . ,cn)].

Note that this property roughly states that the Hamming weight of a polynomial of a linear combi-
nation of polynomials of the form (x1 − 1)i1, . . . (xn − 1)in is at least the Hamming weight of a minimal
term (in the lexicographical order of exponents).

Acknowledgment: The research of the first author was partly supported by the Spanish grants
MTM2007-64704 and MTM2010-21580-C02-02. He was also funded by the Vernon Wilson Endowed
Chair at Eastern Kentucky University during his sabbatical leave.

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83



Monomial-like codes

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84


	Introduction
	The Ambient Space
	Monomial-like codes
	Duality and the Hasse derivative
	A generalisation of the weight retaining property
	References