Fukui, Masaharu

Maruta, Tatsuya

Kyoto University Research Information Repository

Title Optimal linear codes over $GF$(5)(Semigroups, FormalLanguages and Combinatorics on Words)Author(s)Fukui, Masaharu; Maruta, TatsuyaCitation数理解析研究所講究録 (1995), 910: 5-13Issue Date1995-05URL http://hdl.handle.net/2433/59545RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityOptimal linear codes over $GF(5)$Masaharu Fbkui ( )The Graduate School of MathematicsMeijo UniversityandTatsuya Maruta $(\lambda \mathrm{f}\mathrm{f}\mathrm{l} \mathrm{g}-\oplus^{\backslash })$Meijo University, Junior College Division1 IntroductionLet $GF(q)$ denote the Galois field of $q$ elements and let $V(n, q)$ denote the row vectorspace of ordered $\mathrm{n}$-tuples with entries in $GF(q)$ . The (Hamming) distance $d(x, y)$ betweentwo row vectors $x$ and $y$ is definded to be the number of co–ordinate places in which theydiffer. The weight $wt(x)$ of a row vector $x$ is definded to be the number of non-zero entriesof $x$ . Note that$wt(x)=d(_{X}, \mathrm{o})$ .A linear $[n, k]_{q}$-code $C$ is a $k$-dimensional subspace of $V(n, q)$ . The row vectors of $C$are called codewords. The minimum $d\dot{u}$tance of $C$ is the smallest value of the distancesbetween distinct codewords. A linear $[n, k]_{q}$-code with the minimum distance $d$ is calledlinear $[n, k, d]_{q}$-code. We deal with only linear codes, so that is called simply an $[n, k]_{q}$-codeor an [$n,$ $k,$ $d\rfloor_{q}$-code. A generator matrix of an $[n, k]_{q}$-code $C$ is a $k\cross n$ matrix whose $k$ rowvectors form a basis of $C$ .Next we explain about the error correcting of codewords. Suppose that some noisesinvade to a codeword $c$ , transmitted as a message, and that $c$ changed $d$ . Then thefollowing is well known: $d$ can be corrected to $c$ if at most $\mathrm{L}\frac{d-1}{2}\rfloor$ errors occur, where $\lfloor x\rfloor$denotes the largest integer smaller than or equal to $x$ .A good code will have small $n$ (for fast transmission of messages), large $k$ (to enabletransmission of a wide variety of messages) and large $d$ (to correct many errors). Now weconsider the following Problem:910 1995 5-13 5Problem.Optimize one of the parameters $n,$ $k$ and $d$ for given values of the other two.In particular, the problem of optimizing $n$ , i.e. finding the smallest value of $n$ for whichthere exists an $[n, k, d]_{q}$-code for given $k,$ $d$ (we denote the value by $n_{q}(k,$ $d)$ ), is the mostnatural, because later the Griesmer bound provides an important lower bound on $n_{q}(k, d)$ .An [$n_{q}(k, d),$ $k,$ $d\rfloor q$-code is called optimal.Set $g_{q}(k, d)= \sum_{i=0}^{k-1}\mathrm{r}d/q^{i}\rceil$ , where $\lceil x\rceil$ denotes the smallest integer larger than or equal to$x$ . The following theorem is well known:Theorem 1.1. (The Griesmer bound)$n_{q}(k, d)\geq g_{q}(k, d)$ .Theorem 1.2. (R.Hill [4])For given $q$ and $k$ , the Griesmer bound is attaind if $d\geq(k-2)q^{k-}1+1$ .By Theorem 1.2, for a fixed dimension $k,$ $n_{q}(k, d)$ is equal to $g_{q}(k, d)$ for all sufficientlylarge values of $d$ . So the problem of finding $n_{q}(k, d)$ is a finite one. The values of $n_{q}(k, d)$are already known for the following $k(\leq 4),$ $q$ :(i) $k\leq 2$ for all $q$ and $d$ ,(ii) $k=3,$ $q\leq 9$ for all $d$ ,(iii) $k=4,$ $q\leq 4$ for all $d$ ,(iv) $k=4,$ $q=5$ for all but 54 values of $d$ ( $\mathrm{c}.\mathrm{f}$ . I.G.Boukliev and $\mathrm{S}.\mathrm{N}$ .Kapralov [1]).We resolved eight of these fifty four cases in (iv), and we improved one of the remainingones. Table 1 is arranged to put contents of [1] and updated values or bounds together.Details are in Section 3.In Section 3 we only consider optimal codes for $q–5$ and $k=4$ . Note that for $k\leq 2$ ,$n_{5}(k, d)=g_{5}(k, d)$ for all $d$ by Theorem 1.2. For $k=3$, the values of $n_{5}(3, d)$ are as follows( $c.\mathrm{f}$ . R.Hill [4]):$n_{5}(3, d)=\{$$g_{5}(3, d)+1$ for $d=5,9,10,13,14,15$ ,$g_{5}(3, d)$ for other values of $d$ .In Section 2 we give some preliminary results.62 PreriminariesTheorem 2.1. (T.Maruta [7])For $q\geq 4$ there $d$ose not exist an $[n, 4, n+1-q^{2}]_{q}$-code with $n=\lceil q^{3}-q-\sqrt{q}-21$ .Lemma 2.2. ( $\mathrm{P}.\mathrm{P}$ . Greenough and R.Hill [3])(i) $n_{q}(k, d)\leq n_{q}(k, d+1)-1$ , (ii) $n_{q}(k, d)\geq n_{q}(k, d-1)+1$ .Definition.Let $C$ be an $[n, k]_{q}$-code. The dual code of $C$ , denoted by $C^{\perp}$ , is given by$C^{\perp}=$ { $v\in V(n,$ $q)|(v,$ $c)=0$ for all $c\in C$},where $(x, y)$ is the inner product as usual.Theorem 2.3. (The MacWilliams $\mathrm{i}$dentities)Let $C$ be an $[n, k]_{q}$-code. Let $A_{i}$ and $B_{i}$ denote the number of codewords of weight $i$ in$C$ and in the dual code $C^{\perp}$ respectively. Then the $A_{i}’ \mathrm{s}$ and $B_{i}’ \mathrm{s}$ satisfy$j= \sum_{0}^{n-\mathrm{r}}A_{j}=q^{k}-t\sum_{j=0}^{t}B_{j}$for $t=0,1,$ $\ldots$ , $n$ .Definition.Let $G$ be a generator matrix of a linear $[n, k, d]_{q}$-code $C$ . Then the residual code of $C$with respect to a codeword $c$ , denoted by ${\rm Res}(C, c)$ , is the code generated by the restrictionof $G$ to the columns where $c$ has a zero entry.Theorem 2.4. (R.Hill [4])Suppose $C$ is an $[n, k, d]_{q}$-code and suppose $c\in C$ has weight $w$ , where $d> \frac{w(q-1)}{q}$ . Then${\rm Res}(C, c)$ is an $[n-w, k-1, d\mathrm{o}]_{q}$-code with $d^{\mathrm{O}} \geq d-w+\lceil\frac{w}{q}\rceil$ .Definition.A linear code is called projective if no two columns of a generator matrix are linearlydependent.Theorem 2.5. (R.Hill [4])Suppose $d\leq q^{k-1}$ and that $C$ is an $[n, k, d]_{q}$-code which attains the Griesmer bound.Then $C$ is projective.7Note that the columns of a generator matrix of a projective $[n, k]_{q}$-code may be regardedas distinct points of the projective space $PG(k-1, q)$ .Lemma 2.6. (R.Hill [4])There exists a projective $[n, k, d]_{q}$-code if and only if there exists an $n$-set $L$ of $PG(k-1, q)$such that $|L\cap\Pi|\leq n-d$ for any hyperplane II of $PG(k-1, q)$ and that equality hol $d\mathrm{s}$for some hyperplane.In Lemlna 2.6, take the $n$ columns of a generator matrix as an $n$-set $L$ , then it is $\mathrm{e}\mathrm{a}s\mathrm{y}$ toverify that this set satisfies the condition in Lelnma 2.6. Consequently when we considerproblems with respect to a code, we may regard the column vectors of a generator matrixof the code as points of the corresponding projective space.Definition.For a given [$n,$ $k,$ $d\rfloor_{q}$-code $C$ , an $i$-line (resp. an $i$-plane) is aline (resp. a plane) containingexactly $i$ points of $C$ . Denoted by $a_{i}$ the number of planes $\Pi_{i}$ of $PG(k-1, q)$ with $|C\cap\Pi_{i}|=$$i$ .Let $C$ be an $[n, k, d]_{q}$-code which attains the Griesmer bound with $d\leq q^{k-1}$ . By Theorem2.5 $C$ is projective. Then, by Theorem 2.3 We have the following equalities:$. \sum_{1=0}^{n-}d$ $a:= \frac{q^{k}-1}{q-1}$ ,$n- \sum_{i=1}^{d}iai=n\frac{q^{k-1}-1}{q-1}$ ,$, \sum_{2i=}^{\iota-d}i(i-1)a_{i}=n(n-1)\frac{q^{k-2}-1}{q-1}$ .At the following two lemmas we set $k=4$, and we assume that any plane $\Pi$ of $PG(3, q)$meets $C$ in at most $m$ points.Lemma 2.7.If there is a $t$-line, then we have $t \leq\frac{1}{q}\{m(q+1)-|C|\}$ .Lemma 2.8.$a_{i}>1$ implies $i \geq\frac{1}{2}\{|C|-m(q-1)\}$ .83Optimal linear codes over $GF(5)$ with $\mathrm{k}=4$In this section, we explain about Table 1 and our new results. Table 1 is the updatedbest bounds or values of $n_{5}(4, d)$ . For comparison, we also list the values of $g_{5}(4, d)$ . Forall $d$ not listed, in Table 1, it is already known that the Griesmer bound is attained. In thetable, the values labeled $a$ are due to $\mathrm{I}.\mathrm{G}$ .Boukliev [2], labeled $b$ are our new results, andothers are given by $\mathrm{I}.\mathrm{G}$ .Boukliev and $\mathrm{S}.\mathrm{N}$ .Kapralov [1]. Note that the lower bounds aregiven either by the Griesmer bound (Theorem 1.1) or by proving the nonexistence of codesattaining the Griesmer bound and applying Lelnma 2.2. On the other hand, the upperbounds are given by constructing codes with suitable parameters.Next we prove some of our new results.Theorem 3.1.There exist no codes which have the following paranieters:(i) $[44, 4, 34]_{5}$ , (ii) $[50, 4, 39]_{5}$ , (iii) $[106, 4, 84]5$ , (iv) $[110, 4, 87]_{5}$ ,(v) $[115, 4, 91]_{5}$ .Proof. (v) follows from Theorem 2.1. We only prove (ii) here, because the proof of (ii) iscomparatively easy. The other results are also proved similarly making use of the conceptof projective geometry.Let $C$ be a $[50, 4, 39]_{5}$-code. By Theorem 2.5 $C$ is projective. By Theorem 2.4 $A_{i}>0$implies $i\in$ {0,39,40,44,45,49,50}. So, $a_{i}>0$ implies $i\in$ {11,10,6,5,1,0}. Now we have thefollowing Lemma:Lemma 3.2.(i) $|C\cap\Pi|\leq 11$ , for any plane $\Pi$ of $PG(3,5)$ ,(ii) There are no 4-, 5- and 6-1ine,(iii) $a_{0}=0$ ,(iv) $a_{1}=0$ .Proof. (i): This follows from Lemma 2.6.(ii): This follows from Lemma 2.7.(iii): By Lemma 2.8, note that $a_{0}>0$ implies $a_{0}=1$ . Suppose $a_{0}=1$ . Let $L$ be aline containd in the $0$-plane, and consider the six planes containing $L$ . Then from (i) and$|C|=11\cross 4+6$ , we cannot have a 1- and 5–plan$e$ . Hence $a_{1}=a_{5}=0$ . Then we have theunique solution$a_{0}=1$ , $a_{6}= \frac{75}{2}$ , $a_{10}=- \frac{65}{2},$ $a_{11}=150$ ,9which contradicts that $a_{\mathfrak{i}}’ \mathrm{s}$ must be non-negative integers.(iv): Note that $a_{1}>0$ implies $a_{1}=1$ . Suppose $a_{1}=1$ , then we have the unique solution$a_{1}=1$ , $a_{5}=25,$ $a_{6}=1$ , $a_{10}=1,$ $a_{11}=128$ .Let $\Pi_{5}$ and $\Pi_{10}$ be a 5-plane and the 10–plane respectively, and let $L’$ be the int$e$rsectionof $\Pi_{5}$ and $\Pi_{10}$ . Set $C’=C\backslash L’$ . Consider the six planes containing $L’$ . If $L’$ is an i-line,then we have $|C’|=35+i$ for $i=0,1,2,3$, from (ii). But we cannot find the remainingfour planes containing $L’$ from the unique solution. $\square$By Lemma $3.2(\mathrm{i}\mathrm{i}\mathrm{i}),(\mathrm{i}\mathrm{v}),$ $a_{i}>0$ implies $i\in\{11,10,6,5\}$ . Then the MacWilliams identitiesyield a contradiction. $\square$RemarkNote that there exist no $[45, 4, 35]_{5^{-}},$ $[51,4,40]5^{-},$ $[111,4,88]_{5}-$ and $[116, 4, 92]_{5}$ -codes byLemma 2.2 and Theorem 3.1.For the unresolved cases of $k=4$ an$dq=5$ , we can show that optimal codes with thefollowing parameters, if they exist, must have th$e$ unique weight distributions indicated:a $[39, 4, 30]_{5}$-code: $A_{30}=468,$ $A_{35}=156$ .a $[70, 4, 55]_{5}$-code: $A_{55}=512,$ $A_{60}=88$ , $A_{65}=24$ .If there exist both codes with above parameters, then we can result $n_{5}(4,28)=37$ ,$n_{5}(4,29)=38,$ $n_{5}(4,52)=67,$ $n_{5}(4,53)=68$ , and $n_{5}(4,54)=69$ .10Table 1. Values and bounds for $n_{5}(4, d)$ .$dg_{5}(4, d)$ $n_{5}(4, d)$$1$ 4 42 5 53 6 64 7 85 8 96 10 107 11 118 12 129 13 1410 14 1511 16 1612 17 1813 18 1914 19 2015 20 2116 22 2217 23 2318 24 2419 25 2520 26 2621 28 2922 29 3023 30 3124 31 3225 32 33–3426 35 3527 36 $36^{a}$$28$ 37 37–3829 38 38–3930 39 39–4031 41 41–42$dg_{5}(4, d)$ $n_{5}(4, d)$$32$ 42 42–4333 43 43–4434 44 $45^{b}$$35$ 45 $46^{b}-47$$36$ 47 47–4837 48 48–4938 49 49–5039 50 $51^{b}$$40$ 51 $52^{b}$$41$ 53 5442 54 5543 55 $56^{a}$$44$ 56 $57^{a}$$45$ 57 $58^{a}$$46$ 59 6047 60 6148 61 6249 62 6350 63 6451 66 6652 67 67–6853 68 68–6954 69 69–7055 70 70–7156 72 7257 73 7358 74 7459 75 7560 76 7661 78 7962 79 801163 80 8164 81 8297 122 12298 123 12365 82 8366 84 8567 $- 85$ 8668 86 $87^{a}$$69$ 87 $88^{a}$$70$ 88 $89^{a}$$71$ 90 9172 91 9273 92 9374 93 9475 94 9576 97 9777 98 9878 99 9979 100 10080 101 10181 103 103–10482 104 104–10583 105 105–10684 106 $107^{b}$$85$ 107 10886 109 109–11087 110 $111^{b}$$88$ 111 $112^{b}$$89$ 112 11390 113 11491 115 $116^{b}$$92$ 116 $117^{b}$$93$ 117 11894 118 11995 119 12096 121 121146 184 $184-185^{a}$$147$ 185 $185-186^{a}$$148$ 186 $186-187^{a}$$149$ 187 $187-188^{a}$$150$ 188 $188-189^{a}$$151$ 191 191152 192 192153 193 193154 194 194155 195 195156 197 197157 198 198158 199 199159 200 200160 201 201161 203 203–204162 204 204–205163 205 205–206164 206 206–207165 207 207–208166 209 209–210167 210 210–211168 211 211–212169 212 212–213170 213 213–214171 215 215–216172 216 216–217173 217 217–218174 218 218–219175 219 219–22012References[1] I.G.Boukliev and S.N.Kapralov: Optimal linear codes of dimension 4 over $F_{5}$ , preprint.[2] I.G.Boukliev: private communication.[3] P.P.Greenough and R.Hill: Optimal linear codes over $GF(4)$ , Discrete Math. 125,187-199.[4] R.Hill: Optimal linear codes, in: C.Mitchell ed., Proc. 2nd IMA Conf. on Cryptogra-phy and Coding, Oxford Univ. Press, Oxford 1992, 75-104.[5] I.Landgev, T.Maruta and R.Hill: On the nonexistence of quaternary [51,4,34] codes,preprint.[6] F.J.MacWilliams and N.J.A.Sloane: The Theory of Error-Correcting Codes, North-Holland Mathematical Library Vol.16, Ammsterdam, 1977.[7] T.Maruta: On the non-existence of linear codes attaining the Griesmer bound,preprint.13

Optimal linear codes over $GF$(5)(Semigroups, Formal Languages and Combinatorics on Words)

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Optimal linear codes over $GF$ (5)(Semigroups, Formal Languages and Combinatorics on Words)

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Optimal linear codes over GFGFGF(5)(Semigroups, Formal Languages and Combinatorics on Words)

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Optimal linear codes over $GF$ (5)(Semigroups, Formal Languages and Combinatorics on Words)