Betsumiya, Koichi

Kyoto University Research Information Repository

Title Special Self-Dual Codes over $\mathbb{F}_4$(Theory andApplications of Combinatorial Designs with Related Field)Author(s)Betsumiya, KoichiCitation数理解析研究所講究録 (2006), 1465: 119-125Issue Date2006-01URL http://hdl.handle.net/2433/48021RightType Departmental Bulletin PaperTextversionpublisherKyoto University119Special Self-Dual Codes over $\mathrm{F}_{4}$*(Koichi Betsumiya)Department of Business Information SciencesJobu University1 Statement of the resultThe notion “special self-dual codes” in the title of this proceeding is just defined for thisconference temporarily not common terminology. In this proceeding we call a self-dualcode of length 16 over F4 special if the Frobenius automorphism does not preserve thecomplete weight enumerator of the code.Our purpose of this proceeding is to enumerate the special codes. In order to dothis, we give a classification of Type I codes of length 16 over F4 by means of Munemasa’smethod [3], Cor sequently we find that there exist precisely 9858 codes up to permutation-equivalence. 2948 codes of them are the special codes.In order to obtain this result, we calculated by means of the software package MAGMAon an AMD Opteron 242 $(64\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}1.6\mathrm{M}\mathrm{H}\mathrm{z})$ machine Linux-operated. We needed about 2hours for the total calculation.We organize this proceeding as following. First, we give necessary terminology inSection 2. In Section 3 we introduce a background of this work. In Section 6 we givethe main result of this proceeding – an enumeration of special codes. In Section 4 andSection 5, we give a classification $[16, 7]$-self-orthogonal codes and Type 1 code of length16 respectively to show the main result.2 Notation and DefinitionsLet $\mathrm{F}_{4}$ be the finite field constructed by 4 elements $\{0, 1, \omega, \omega^{2}\}$ where $\omega^{2}+\omega$ $+1=0$ .A linear code of length $n$ is a subspace of the vector space $\mathrm{F}_{4}^{n}$ . A [$n,$ $k_{\rfloor}^{\rceil}$-code is a code ofThis work was partially supported by the MEXT; Japan, Grant-in-Aid for Young Scientists (B)(16740023)1465 2006 119-125120length $n$ of dimension $k$ . With respect to a inner product $(, )$ , we define the dual code ofa code $C$ to be$C^{[perp]}=$ { $u\in \mathrm{F}_{4}^{n}|$ ( $u$ , $v)=0$ for all $v\in C$ }.If $C=C^{[perp]}$ , $C$ is said to be self-dual If $C\subset C^{[perp]}$ , $C$ is said to be self-orthogonal. Throughthis proceeding we fix the inner product to be the Euclidean inner product$(u, v)$ $= \sum_{\mathrm{i}=1}^{n}u_{i}v_{i}$ .A vector $(c_{1}, c_{2}, \ldots, c_{n})$ satisfies $\Sigma_{?=1}^{n}c_{\overline{I}}=0$ and $\Sigma_{i<j}c_{i}c_{j}=0$ is called doubly-even.For a self-dual code $C$ , if every codeword $c\in C$ is doubly-even, $C$ is called Type $II$ ordoubly-even, otherwise $C$ is called Type I or singly-even.When a code $D$ is obtained by permutating coordinates of a code $C$ , we call $D$ ispermutation-equivalent to $C$ . When a code $D$ is obtained by perm utating coordinates andby the Frobenius automorphism operating entrywise of a code $C$ , we call $D$ is equivalentto $C$ where the Frobenius automorphism is a map defined by $xrightarrow x^{2}$ . Let $C$ be a code,$S_{n}$ be the symmetric group of degree $n$ and $\mathrm{G}\mathrm{a}1_{\mathrm{F}_{4}/\mathrm{F}_{2}}$ be the Galois group of F4 over $\mathrm{F}_{2}$generated by the Frobenius automorphism. The groupsPAut(C) $:=$ {a $\in S_{n}|C=C^{\sigma}$ }andAut(C) $:=\{\sigma\in S_{n}\cross \mathrm{G}\mathrm{a}1_{\mathrm{F}_{4}/\mathrm{F}_{2}}|C=C’\}$are called the permutation-automorphism group and the automorphism group of $C$ respec-tively.The complete weight enumerator of a code $C$ is a polynomial in $\mathbb{C}$ [ $y_{\alpha}|$ a $\in \mathrm{F}_{4}$] definedby$\mathrm{c}\mathrm{w}\mathrm{e}_{C}(y_{0}, y_{1}, y_{\omega}, y_{\alpha r^{2}})=\sum_{\mathrm{c}\in C}\prod_{i=1}^{n}y_{c_{i}}$.Now we define that a special code is a self-dual code of length 16 over $\mathrm{F}_{4}$ of whichcomplete weight enumerator is not preserved by the Frobenius automorphism.3 Background of this WorkLet $M$ , $g_{1}$ , $g_{2}$ and $f$ be ring homomorphisms on the polynomial ring $\mathbb{C}$ [ $y_{\alpha}|$ a $\in \mathrm{F}_{4}$ ] definedby $M(y_{\alpha}):=(1/2)\Sigma_{\beta\in \mathrm{F}_{4}}(-1)^{\mathrm{T}\mathrm{r}(\alpha\beta)}y_{\beta}$, $g_{1}(y_{\alpha}):=y_{\alpha+1\}}g_{2}(y_{\alpha})$ $:=y_{\alpha\omega}$ and $f(y_{\alpha})$ $:=y_{\alpha^{2}}$ .121These homomorphisms are described by matrices indexed by $\mathrm{F}_{4}=\{0,1, \omega, \omega^{2}\}$ as$\frac{1}{2}\{$1111$\backslash$11 -1 -11 -I -1 11 -1 1 -1,$\{$01 0 $0^{\backslash }$$1$ 0 0 00001001 $0_{/}$$\{$100 $0^{\backslash }$0001010000 1 0,and $(\begin{array}{llll}1 0 \mathrm{O} 00 \mathrm{l} \mathrm{O} 00 0 0 10 0 1 0\end{array})$respectively. Let $G$ and $H$ be groups generated by $\{M, g_{1}, g_{2}, f\}$ and $\{M, g_{1}, g_{2}\}$ respec-tively.It is known that the invariant ring $\mathbb{C}[y_{\alpha}|\mathrm{a}\in \mathrm{F}_{4}]^{H}$ coincides with the ring generatedby the complete weight enumerators of self-dual codes $(\mathrm{c}.\mathrm{f}. [2, 4])$ . The Molien series of$H$ is$\sum_{i=0}^{\infty}\dim_{i}(\mathbb{C}[y_{\alpha}|\mathrm{a}\in \mathrm{F}_{4}]^{H})t^{i}=\frac{1+t^{16}}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{8})}$.The other hand, the Molien series $G$ is$\sum_{i=0}^{\infty}\dim_{i}(\mathbb{C}[y_{\alpha}|\alpha \in \mathrm{F}_{4}]^{G})t^{2}=\frac{1}{(1-t^{2})(1-t^{4})(1-t^{6})(1-t^{8})}$ .These Molien series indicate that $\mathbb{C}[y_{\alpha}|\mathrm{a}\in \mathrm{F}_{4}]^{G}$ is a polynomial ring and$\mathbb{C}[y_{\alpha}|\mathrm{a}\in \mathrm{F}_{4}]^{H}=\mathbb{C}[y_{\alpha}|\mathrm{a}\in \mathrm{F}_{4}]^{G}\oplus \mathbb{C}[y_{\alpha}|\mathrm{a}\in \mathrm{F}_{4}]^{G}\phi_{16}$where $\phi_{16}$ is a complete weight enumerator of special code. An example of such code wasconstructed in [2] as a double circulant code $C_{16}$ .Prom the facts above, it is natural to appear some questions:$\bullet$ there exists the other example or not,$\bullet$ there exists a certain standard code like other generators or not.In order to answer these questions, we will give a classification of such codes in thefollowing sections.4 Enumeration of [16,$7]$ -Self-Orthogonal Codes Containing 1In this section we will give a classification of $[16, 7]$ -self-orthogonal codes which containall-one vector by means of the classification of Type II codes of length 16 in [1]122Theorem 4.1 There exist 3611 $[16, 7]$ -self-orthogonal codes containing 1 up to permutation-equivalence. For 1469 of them, the Frobenius map image of the code are pemutation-equivalent of itself. Hence, there exist 2540 $[16, 7]$ -self-orthogonal codes containing 1 upto equivalence.We will explain how to obtain the theorem. Before doing this, we recall the facts asfollowing.Proposition 4.2 ([I]) There precisely exist 48 TyPe II codes of length 16 uP to permutation-equivalence.Proposition 4.3 Any [2k, k $-1]$ -self-orthogonal code containing 1 is contained in pre-cisely two Type II codes of length 2k.Let $\mathrm{C}$ be a Type II code of length 16. By means of the method in [3], we constructall hyperplanes of $\mathrm{C}$ up to equivalence with respect to Aut(C) action. Repeating thecalculation for each Type II code, we obtain distinct 10719 $[16, 7]$ -self-orthogonal codes.To classify these codes up to equivalence, we compute the following invariants of each$[16, 7]$ -self-orthogonal code $\mathrm{H}$ :$\bullet$ The two Type II codes containing $\mathrm{H}$ ,$\bullet$ The complete weight enumerator of $\mathrm{H}$ ,$\bullet$ The invariant twoptdegreeSorted for weight 8 defined in [3].For the first invariant, one of the Type II codes is $\mathrm{C}$ which is the initial Type II codeconstracting H. We obtain the other Type II code $\mathrm{D}$ containing $\mathrm{H}_{-}$ by means of the followingMAGMA script.$\mathrm{K}<\mathrm{a}>:=\mathrm{G}\mathrm{F}(4)$ ;$\mathrm{V}:=\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{t}\mathrm{K}$, 16) ;$\mathrm{U};=\mathrm{U}\mathrm{n}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{d}\mathrm{e}$( $\mathrm{K}$ , i6) ;$\mathrm{H}\mathrm{O}:=\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}(\mathrm{H})$ ;Hd: $=\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}$ (Dual (H));for $\mathrm{v}$ in Transversal (Hd, $\mathrm{H}\mathrm{O}$) doif $\mathrm{v}$ ne 0 and not $\mathrm{U}!\mathrm{v}$ in $\mathrm{C}$ andLW $(\mathrm{a}*\mathrm{v})$ mod 4 eq 0 and $\mathrm{L}\mathrm{W}(\mathrm{a}^{\wedge}2*\mathrm{v})$ mod 4 eq 0 then$\mathrm{D}:=\mathrm{L}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{C}\mathrm{o}\mathrm{d}\mathrm{e}$ (sub$<\mathrm{V}$ $|\mathrm{H}$ , $\mathrm{v}>$);break v;end if;end for;123Here we give an algorithm to obtain twoptdegreeSorted. It is clear that this isinvariant of equivalence class as well as permutation-equivalence class.$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}:arrow-\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}$ ion $(\mathrm{C}, \mathrm{t})$return Inc idenc $\mathrm{e}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}<$ Length (C) $|${ Support (u) : $\mathrm{u}$ in $\mathrm{C}|$ Weight (u) eq $\mathrm{t}$ } $>i$end function;twoptdegreeSort $\mathrm{e}\mathrm{d}:=\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}(\mathrm{C},\mathrm{t})$des: $=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{t}\mathrm{C},\mathrm{t}$ ) ;return Sort $([$$\#${ $\mathrm{b}$ : $\mathrm{b}$ in BlockSet(des) $|$ {Point(des, j) : $\mathrm{j}$ in $\mathrm{x}$ } subset $\mathrm{b}$ }: $\mathrm{x}$ in Subsets $(\{1. .16\}_{;}2)1$ $)$ ;end function;Eventually we obtain 2540 different invariants. Moreover we can verify that the in-variant characterizes inequivalence class of codes by means of mass formula as following.Theorem 4.4 If $n$ is a positive integer divisible by 4, then the number of the $[n, n/2-1]-$self-orthogonal codes containing 1 is equal to$\frac{1}{3}(\prod_{i=1}^{n/2-1}(4^{i}+1)-\prod_{i=0}^{n/2-2}(4^{i}+1))$ $=$ 32563225114005625$=$ $\sum_{c\in c}\frac{n!\cdot 2}{|\mathrm{A}\mathrm{u}\mathrm{t}(C)|}$$=$ $\sum_{c\in \mathrm{C}’}\frac{n!}{|\mathrm{P}\mathrm{A}\mathrm{u}\mathrm{t}(C)|}$ ,where $C$ and $\mathrm{C}’$ are a complete sets of representatives of equivalence classes of $[n, n/2-1]-$self-orthogonal codes and permutation-equivalence classes of $[n, n/2 -1]$ -self-orthogonalcodes respectively.There is no build-in command in MAGMA to calculate our automorphism groups.So we calculate the order of automorpshim groups by means of the following MAGMAscript which is modified the script in [3]. The function auto returns $|$ PAut(C) $|$ and$|$ Aut(C) : PAut $(\mathrm{C})|$ .GoodDist: $=\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $(\mathrm{C}, \mathrm{c}\mathrm{w}\mathrm{e})$ ;$\mathrm{U};=\mathrm{G}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{c}(\mathrm{C})$ ;$\mathrm{m}1:=\mathrm{U}![\mathrm{K}$ ! 1: $\mathrm{i}$ in [1. . Length (U) 11 ;$\mathrm{C}\mathrm{F}:=\mathrm{s}\mathrm{u}\mathrm{b}<\mathrm{U}|$ [ $\mathrm{U}$ ! [ $\mathrm{g}^{\wedge}2$ : $\mathrm{g}$ in Eltseq(C. j) 1 : $\mathrm{j}$ in [1. .Dimension(C)]] $>$ ;mono: $=\mathrm{M}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{s}$ $(\mathrm{c}\mathrm{w}\mathrm{e})$ ;$\mathrm{f}$or $\mathrm{j}$ in [2. . #mono-ll doif Degree (mono $[\mathrm{j}]$ , $\mathrm{y}2$) eq Degree (mono $[\mathrm{j}]$ , $\mathrm{y}3$ ) andMonomialCoefficient ( $\mathrm{c}\mathrm{w}\mathrm{e}$ ,mono $[\mathrm{j}]$ ) gt 72 then$\mathrm{p}\mathrm{o}\mathrm{w}:=<\mathrm{D}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}$ (mono $[\mathrm{j}],\mathrm{v}$) : $\mathrm{v}$ in $[\mathrm{y}0, \mathrm{y}1,\mathrm{y}2]$ $>$ ;break j;end if;end $\mathrm{f}$or;end function;124auto: $=\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}(\mathrm{C}, \mathrm{c}\mathrm{w}\mathrm{e})$ ;pow $:=\mathrm{G}\mathrm{o}\mathrm{o}\mathrm{d}\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}(\mathrm{C}, \mathrm{c}\mathrm{w}\mathrm{e})$ ;SP $:=$ { Support (u) : $\mathrm{u}$ in $\mathrm{C}$ $|$$<\mathrm{L}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}(\mathrm{C})$-Distance $(\mathrm{u},\mathrm{x})$ : $\mathrm{x}$ in [$\mathrm{U}$ ! 0, $\mathrm{m}1$ , $\mathrm{a}*\mathrm{m}\mathrm{l}\mathrm{l}$ $>$ eq $\mathrm{p}\mathrm{o}\mathrm{w}$ ;$\mathrm{D}:=\mathrm{I}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}<$ length (C) $|\mathrm{S}\mathrm{P}>$ .$\mathrm{A}:=\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{G}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{t}\mathrm{D})$ ;orbit: $=\mathrm{C}^{-}\mathrm{A}j$if CF in orbit thenfrob: $=2$ ;elsefrob: $=1$ ;end if;return $<\#\mathrm{A}/\#\mathrm{o}\mathrm{r}\mathrm{b}$it , frob$>$ ;end function;5 Enumeration of Type I Self-Dual CodesIn this section we will give a classification of Type I codes of length 16 by means of theresult above,Theorem 5.1 There exist 9858 Type I codes up to permutation-equivalence. For 2200of them, the Frobenius map image of the code is permutation-equivalent to itself. Hence,there exist 6029 Type I codes up to equivalence.We will explain how to obtain the theorem. Before doing this, we recall the facts asfollowing.Proposition 5.2 Any Type I codes of length 2k contains a unique [2k, k-l] self-orthogonalcode.Proposition 5.3 Any [2fc, k-l] self-orthogonal code is contained in precisely three Type $I$codes of length 2k.Let $\mathrm{H}$ be a $[16, 7]$-self-orthogonal code. As same way to obtain a Type II code containing$\mathrm{H}$ in the previous section, we obtain three Type I code containing H. The orbit of Aut(H)on the Type I codes shows us whether the three of them are equivalent or not. Thereforewe obtain the classification of Type I codes of length 16. The mass formula [4] alsoguarantees the classification is complete.Theorem 5.4 (c.f.$[4])$ if n is a positive integer divisible by 4, then the number of the125Type I codes of length $n$ is equal to$\prod_{i=1}^{n/2-1}(4^{i}+1)-\prod_{i=0}^{n/2-2}(4^{i}+1)$ $=$ 97689675342016875$=$ $\sum_{C\in \mathrm{C}}\frac{n!\cdot 2}{|\mathrm{A}\mathrm{u}\mathrm{t}(C)|}$$=$ $\sum_{C\in C’}\frac{n!}{|\mathrm{P}\mathrm{A}\mathrm{u}\mathrm{t}(C)|}$ ,wAere $\mathrm{C}$ and $\mathrm{C}’$ are a complete sets of representatives of equivalence classes of Type I codesof length $n$ and $pe$ rmutation-equivalence classes of Type I codes of length $n$ respectively.6 Enumeration of Special Self-Dual CodesIt is easy to distinguish special self-dual codes among Type I codes.Theorem 6.1 There precisely exist 2948 special self-dual cod es up to permutation-equivalence.In particular there precisely exist 1474 special self-dual codes up to equivalence.Unfortunately we could not succeed to find some standard one among them like othergenerators.References[1] K. Betsumiya. On the classification of Type II codes over $\mathrm{F}_{2^{r}}$ with binary length 32.preprint, 2001.[2] K. Betsumiya and Y. Choie. Codes over F4, Jacobi forms and Hilbert-Siegel modularforms over $\mathbb{Q}(\sqrt{5})$ . European J. Combin., 26:629-650, 2005.[3] A. Munemasa. On the enumeration of self-dual codes. In Proceedings of the FourthConference on Algebraic Geometry, Number Theory, Coding Theory and C yptography,University of Tokyo, 2000 pages 69-77, 2000.[4] G. Nebe, E. M. Rains, and N. J. A. Sloane. $\mathit{3}elf$-Dual Code and Invariant Theory.2005. working draft

Special Self-Dual Codes over $\mathbb{F}_4$(Theory and Applications of Combinatorial Designs with Related Field)

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Special Self-Dual Codes over $\mathbb{F}_4$ (Theory and Applications of Combinatorial Designs with Related Field)

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Special Self-Dual Codes over F4\mathbb{F}_4F4​(Theory and Applications of Combinatorial Designs with Related Field)

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Special Self-Dual Codes over $\mathbb{F}_4$ (Theory and Applications of Combinatorial Designs with Related Field)