ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.327391

J. Algebra Comb. Discrete Appl.
5(1) • 1–4

Received: 27 December 2016
Accepted: 6 April 2017

Journal of Algebra Combinatorics Discrete Structures and Applications

Ternary maximal self-orthogonal codes of lengths 21, 22
and 23

Research Article

Makoto Araya, Masaaki Harada, Yuichi Suzuki *

Abstract: We give a classification of ternary maximal self-orthogonal codes of lengths 21, 22 and 23. This
completes a classification of ternary maximal self-orthogonal codes of lengths up to 24.

2010 MSC: 94B05

Keywords: Ternary code, Self-dual code, Self-orthogonal code

1. Introduction

A ternary [n, k] code C is a k-dimensional vector subspace of Fn3 , where F3 denotes the finite field
of order 3. All codes in this note are ternary. The parameters n and k are called the length and the
dimension of C, respectively. The weight of a vector x ∈ Fn3 is the number of non-zero components of
x. A vector of C is a codeword of C. The minimum non-zero weight of all codewords in C is called the
minimum weight of C. Two codes C and C′ are equivalent if there is a (0, 1,−1)-monomial matrix P
with C′ = C · P = {xP | x ∈ C}, and inequivalent otherwise. The automorphism group Aut(C) of C is
the group of all (0, 1,−1)-monomial matrices P with C = C ·P .

The dual code C⊥ of a code C of length n is defined as C⊥ = {x ∈ Fn3 | x · y = 0 for all y ∈ C},
where x · y is the standard inner product. A code C is self-dual if C = C⊥, and C is self-orthogonal if
C ⊂ C⊥. A self-dual code of length n exists if and only if n ≡ 0 (mod 4). A self-orthogonal code C is
maximal if C is the only self-orthogonal code containing C. A self-dual code is automatically maximal.
The dimension of a maximal self-orthogonal code of length n is a constant depending only on n. More
precisely, a maximal self-orthogonal code of length n has dimension (n−1)/2 if n is odd, n/2−1 if n ≡ 2
(mod 4) (see [8]).

Makoto Araya (Corresponding Author); Department of Computer Science, Shizuoka University, Hamamatsu
432–8011, Japan (email: araya@inf.shizuoka.ac.jp).
Masaaki Harada; Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences,

Tohoku University, Sendai 980–8579, Japan (email: mharada@m.tohoku.ac.jp).
Yuichi Suzuki; Hitachi Systems, Ltd., 1–2–1, Osaki, Shinagawa-ku, Tokyo, 141–0032, Japan.

* This author carried out his work at Yamagata University.

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http://orcid.org/0000-0002-9935-038X
http://orcid.org/0000-0002-2748-6456


M. Araya et al. / J. Algebra Comb. Discrete Appl. 5(1) (2018) 1–4

A classification of maximal self-orthogonal codes of lengths up to 12, lengths 13, 14, 15, 16 and lengths
17, 18, 19, 20 was done in [8], [2] and [9], respectively (see [4] for lengths 18 and 19). In this note, we give
a classification of maximal self-orthogonal codes of lengths 21, 22 and 23. The mass formula is used to
verify that our classification is complete. Since a classification of self-dual codes of length 24 was done
in [3], our result completes a classification of maximal self-orthogonal codes of lengths up to 24.

2. Classification results

Let C be a code of length n and let S = {i1, i2, . . . , ij} be a subset of {1, 2, . . . , n}. A shortened
code of C is the set by selecting only the codewords of C having zeros in each of the coordinate positions
i1, i2, . . . , ij and deleting these components. Throughout this note, we denote the code by C(S). All
maximal self-orthogonal codes of lengths 4m + 1, 4m + 2, 4m + 3 can be obtained from self-dual codes of
length 4m + 4 as shortened codes (see [2]).

For length 24, there are 338 inequivalent self-dual codes, two of which have minimum weight 9, 166
of which have minimum weight 6 and 170 of which have minimum weight 3 [3] and [7]. From the 338
self-dual codes C of length 24, we found maximal self-orthogonal codes of lengths 23 and 22, which must
be checked further for equivalences, as shortened codes C(S) by considering all sets S with |S| = 1 and
2, respectively. This computer calculation was done by using the Magma [1] function ShortenCode.
Then we determined the equivalence or inequivalence of two codes among the maximal self-orthogonal
codes. This calculation was done by the Magma function IsIsomorphic. Then we have 13625 and 2005
inequivalent maximal self-orthogonal codes of lengths 22 and 23, respectively. Note that the dimensions of
maximal self-orthogonal codes of lengths 21 and 22 are 10. The 126 codes among the 13625 maximal self-
orthogonal codes of length 22 have a zero coordinate. Hence, 216 inequivalent maximal self-orthogonal
codes of length 21 are obtained, as shortened codes. We denote by C(n, d) the set of the inequivalent
maximal self-orthogonal codes of length n and minimum weight d for (n, d) = (21, 3), (21, 6), (22, 3),
(22, 6), (22, 9), (23, 3), (23, 6) and (23, 9). In addition, we define subsets of C(n, d):

C(n, d, d′) = {C ∈C(n, d) | d(C⊥) = d′},

where d(C) denotes the minimum weight of C. The numbers |C(n, d, d′)| are listed in Table 1.
As a check, in order to verify that C(n, d) contains no pair of equivalent codes for the above (n, d), we

employed the following method obtained by applying the method given in [6, Section 2]. Let C be a code
of length n. Suppose that t is a positive integer such that the codewords of weight t generate C. Let At
denote the number of codewords of weight t in C. We expand each codeword of C into a binary vector of
length 2n by mapping the elements 0, 1 and 2 of F3 to the binary vectors (0, 0), (0, 1) and (1, 0), respec-
tively. In this way, we have an At × 2n binary matrix M(C, t) composed of the binary vectors obtained
from the At codewords of weight t in C. Then, from M(C, t), we have an incidence structure D(C, t)
having 2n points. This calculation was done by using the Magma function IncidenceStructure. If C
and C′ are equivalent, then D(C, t) and D(C′, t) are isomorphic. By the Magma function IsIsomorphic,
we verified that the incidence structures D(C, t) are non-isomorphic for the above (n, d). This shows that
C(n, d) contains no pair of equivalent codes for the above (n, d).

The number of distinct maximal self-orthogonal codes of length n is known [8] as:

N(n) =

{∏(n−1)/2
i=1 (3

i + 1) if n is odd,∏n/2
i=2(3

i + 1) if n ≡ 2 (mod 4).

We calculated the following values:

T(n, d, d′) =
∑

C∈C(n,d,d′)

2n ·n!
|Aut(C)|

.

The results are listed in Table 1. The automorphism groups of the codes were calculated by the Magma
function AutomorphismGroup. We remark that the automorphism group of a code C is isomorphic to the

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M. Araya et al. / J. Algebra Comb. Discrete Appl. 5(1) (2018) 1–4

Table 1. |C(n, d, d′)| and T(n, d, d′).

(n, d, d′) |C(n, d, d′)| T(n, d, d′)
(21, 3, 1) 18 37261233666612695040000

(21, 3, 3) 129 22666803510606607679488000

(21, 6, 1) 6 156912620925725599334400

(21, 6, 4) 59 221566090068991210527129600

(21, 6, 6) 4 28572125748609278803968000

(22, 3, 1) 147 499079550803678108594176000

(22, 3, 2) 671 8999173098190687835078656000

(22, 3, 3) 3606 397450658156464202444177408000

(22, 6, 1) 69 5504766786817393746876825600

(22, 6, 2) 458 116255553756749319466332979200

(22, 6, 4) 6528 8198363298466655101459523174400

(22, 6, 5) 2142 3362889158614819168464981196800

(22, 9, 7) 4 353580056139039825199104000

(23, 3, 2) 153 23004306466349702422944153600

(23, 3, 3) 728 1838692744522339728778225254400

(23, 6, 2) 63 253139874407411695203070771200

(23, 6, 5) 1059 46245009828325897079698017484800

(23, 9, 8) 2 1414320224556159300796416000

stabilizer of {{1, 2},{3, 4}, . . . ,{2n−1, 2n}} inside of the automorphism group of the incidence structure
D(C, t). In order to verify the correctness of the above calculations of the automorphism groups, we also
calculated the stabilizers for D(C, t). This was done by the Magma function Stabilizer. Finally, as a
check, we verified the mass formula:

N(21) =T(21, 3, 1) + T(21, 3, 3) + T(21, 6, 1) + T(21, 6, 4) + T(21, 6, 6),

N(22) =T(22, 3, 1) + T(22, 3, 2) + T(22, 3, 3) + T(22, 6, 1)

+ T(22, 6, 2) + T(22, 6, 4) + T(22, 6, 5) + T(22, 9, 7),

N(23) =T(23, 3, 2) + T(23, 3, 3) + T(23, 6, 2) + T(23, 6, 5) + T(23, 9, 8).

The mass formula shows that there is no other maximal self-orthogonal code of lengths 21, 22 and 23.
We summarize a classification of maximal self-orthogonal codes of lengths 21, 22 and 23.

Proposition 2.1. (1) Up to equivalence, there are 216 maximal self-orthogonal codes of length 21, 147
of which have minimum weight 3 and 69 of which have minimum weight 6.

(2) Up to equivalence, there are 13625 maximal self-orthogonal codes of length 22, 4424 of which have
minimum weight 3, 9197 of which have minimum weight 6 and 4 of which have minimum weight 9.

(3) Up to equivalence, there are 2005 maximal self-orthogonal codes of length 23, 881 of which have
minimum weight 3, 1122 of which have minimum weight 6 and 2 of which have minimum weight 9.

Remark 2.2. Generator matrices of all the maximal self-orthogonal codes of lengths 21, 22 and 23 can
be obtained electronically from [5].

Acknowledgment: The first and second authors would like to thank Ken Saito for helpful discus-
sions on the classification method given in [6]. The first author would also like to thank the Graduate
School of Information Sciences, Tohoku University for the hospitality during his visit in August and
December 2016. This work is supported by JSPS KAKENHI Grant Numbers 15H03633 and 15K04976.

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	Introduction
	Classification results
	References