Let nq(k, d) denote the smallest value of n for which an [n, k, d]q code exists for given integers k and d with k ≥ 3, 1 ≤ d ≤ q^(k−1)
and a prime or a prime power q.  The purpose of this note is to show that there exists a series of the functions h3,q, h4,q, ..., hk,q
such that nq(k, d) can be expressed.This research was partially supported by Grant-in-Aid for Scientific Research of Japan
Society for the Promotion of Science under Contract Number 20540129

Hamada, Noboru

Maruta, Tatsuya

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica J. Computing 5 (2011), 199–206NOTE ON AN IMPROVEMENT OF THE GRIESMERBOUND FOR q-ARY LINEAR CODESNoboru Hamada, Tatsuya Maruta∗Abstract. Let nq(k, d) denote the smallest value of n for which an [n, k, d]qcode exists for given integers k and d with k ≥ 3, 1 ≤ d ≤ qk−1 and a primeor a prime power q. The purpose of this note is to show that there exists aseries of the functions h3,q, h4,q, . . . , hk,q such that nq(k, d) can be expressedas nq(k, d) =k−1∑i=0⌈d/qi⌉ +k∑j=3hj,q(ek−j , ek−j+1, . . . , ek−2) for some ordered(k − 1)-tuple (e0, e1, . . . , ek−2) with 0 ≤ e0, e1, . . . , ek−2 ≤ q − 1 satisfyingd = qk−1 −k−2∑i=0eiqi.1. Introduction. Let Fnq denote the vector space of n-tuples over Fq,the field of q elements, where n is an integer ≥ 4 and q is a prime or a prime power.A q-ary linear code C of length n and dimension k, called an [n, k]q code, is ak-dimensional subspace of Fnq , where n > k ≥ 3. An [n, k]q code C with minimumACM Computing Classification System (1998): E.4.Key words: linear codes, Griesmer bound.*This research was partially supported by Grant-in-Aid for Scientific Research of JapanSociety for the Promotion of Science under Contract Number 20540129.200 Noboru Hamada, Tatsuya MarutaHamming distance d is referred to as an [n, k, d]q code. Let G = [gT1 ,gT2 , · · · ,gTn ]be a k×n generator matrix of an [n, k, d]q code C with g1, · · · ,gn ∈ Fkq , where gTdenotes the transpose of the vector g. If there is no zero vector in {g1, · · · ,gn},an [n, k, d]q code C is called a nontrivial code. A fundamental problem in codingtheory is to solve the following problem.Problem 1. Find the smallest value of n, denoted by nq(k, d), for whichan [n, k, d]q code exists for given integers q, k, d.An [n, k, d]q code is called optimal if n = nq(k, d). There is a lower boundon nq(k, d) called the Griesmer bound [2], [5]:nq(k, d) ≥ gq(k, d) :=k−1∑i=0⌈dqi⌉,where ⌈x⌉ denotes the smallest integer greater than or equal to x. A [gq(k, d), k, d]qcode is called a Griesmer code. In this note, we consider the case k ≥ 3, q ≥ 3and 1 ≤ d ≤ qk−1. In this case, d and gq(k, d) can be expressed as follows:d = qk−1 −k−2∑i=0eiqi,(1.1)gq(k, d) = θk−1 −k−2∑i=0eiθi(1.2)using some ordered (k−1)-tuple (e0, e1, . . . , ek−2) in E(k, q), where E(k, q) is theset of all ordered (k − 1)-tuple (e0, e1, . . . , ek−2) such that 0 ≤ ei ≤ q − 1 andθi = (qi+1 − 1)/(q − 1) for 0 ≤ i ≤ k − 2. In the special case k = 3, d can beexpressed as follows:d = q2 − (e0 + e1q).Note that (1.2) shows that gq(k, d) is a function of k, e0, e1, . . . , ek−2 and q. Now,we define the Hamada’s function hk,q(e0, e1, . . . , ek−2) for k ≥ 3 as follows:h3,q(e0, e1) = nq(3, d) − gq(3, d),(1.3)hk,q(e0, e1, · · · , ek−2) = nq(k, d)− gq(k, d)(1.4)−k−1∑j=3hj,q(ek−j , ek−j+1, · · · , ek−2)Note on an Improvement of the Griesmer Bound for q-ary Linear Codes 201for k ≥ 4, where d is uniquely determined from k, q and (e0, e1, . . . , ek−2) ∈E(k, q) by (1.1).Theorem 1.1 For given q ≥ 3, k ≥ 3 and for any (e0, e1, · · · , ek−2) ∈E(k, q),hk,q(e0, e1, · · · , ek−2) ≥ 0(1.5)holds and nq(k, d) for d satisfying (1.1) can be expressed asnq(k, d) = gq(k, d) +k∑j=3hj,q(ek−j , ek−j+1, · · · , ek−2).(1.6)Remark 1.2. The formula (1.6), called the Hamada’s formula, showsthat there exists a series of Hamada’s functions h3,q, h4,q, . . . , hk,q such thatnq(k, d) can be expressed as (1.6), where hj,q = hj,q(ek−j, ek−j+1, . . . , ek−2).Hence Problem 1 for 1 ≤ d ≤ qk−1 is equivalent to the following problem.Problem 2. Find the Hamada’s function hk,q = hk,q(e0, e1, · · · , ek−2)such that nq(k, d) can be expressed as (1.6) for given integers k ≥ 3 and q ≥ 3.Example 1.3 [cf. Appendix]. For q = 3 and 3 ≤ k ≤ 5, hk,3(e0, e1, . . . ,ek−2) is given by(1) hk,3(e0, e1, · · · , ek−2) = 0 or 1 for all (e0, e1, · · · , ek−2) ∈ E(k, 3),(2) h3,3(e0, e1) = 1 if and only if (e0, e1) = (0, 2),(3) h4,3(e0, e1, e2) = 1 if and only if (e0, e1, e2) ∈ {(0, 2, 2), (2, 1, 1), (1, 1, 1),(0, 1, 1)},(4) h5,3(e0, e1, e2, e3) = 1 if and only if (e0, e1, e2, e3) ∈ {(0, 2, 2, 2), (2, 1, 1, 2),(1, 1, 1, 2), (0, 1, 1, 2), (2, 0, 1, 2), (1, 0, 1, 2), (0, 0, 1, 2), (2, 0, 0, 2), (1, 0, 0, 2),(0, 0, 0, 2), (1, 1, 2, 1), (0, 1, 2, 1), (2, 2, 0, 1), (1, 2, 0, 1), (0, 2, 0, 1), (2, 1, 0, 1),(1, 1, 0, 1), (0, 1, 0, 1), (2, 0, 2, 0), (1, 0, 2, 0), (0, 0, 2, 0)}.Example 1.4. In the case q = 4 and 3 ≤ k ≤ 4, hk,4(e0, e1, · · · , ek−2) isgiven by(1) hk,4(e0, e1, · · · , ek−2) = 0 or 1 for all (e0, e1, · · · , ek−2) ∈ E(k, 4),202 Noboru Hamada, Tatsuya Maruta(2) h3,4(e0, e1) = 1 if and only if (e0, e1) ∈ {(1, 2), (0, 2)},(3) h4,4(e0, e1, e2) = 1 if and only if (e0, e1, e2) ∈ {(1, 3, 3), (0, 3, 3), (1, 2, 3),(0, 2, 3), (3, 0, 3), (2, 0, 3), (1, 0, 3), (0, 0, 3), (1, 2, 2), (0, 2, 2), (3, 2, 1),(2, 2, 1), (1, 2, 1), (0, 2, 1), (3, 1, 1), (2, 1, 1), (1, 1, 1), (0, 1, 1)}.Example 1.5. In the case q = 5 and 3 ≤ k ≤ 4, hk,5(e0, e1, · · · , ek−2) isgiven by(1) hk,5(e0, e1, · · · , ek−2) = 0 or 1 for all (e0, e1, · · · , ek−2) ∈ E(k, 5),(2) h3,5(e0, e1) = 1 if and only if (e0, e1) ∈ {(0, 4), (1, 3), (0, 3), (2, 2), (1, 2),(0, 2)},(3) h4,5(e0, e1, e2) = 1 if (e0, e1, e2) ∈ {(1, 4, 4), (0, 4, 4), (1, 3, 4), (0, 3, 4),(3, 2, 4), (2, 2, 4), (1, 2, 4), (0, 2, 4), (0, 0, 4), (4, 3, 3), (3, 3, 3), (2, 3, 3),(1, 3, 3), (0, 3, 3), (4, 2, 3), (3, 2, 3), (2, 2, 3), (1, 2, 3), (0, 2, 3), (2, 3, 1),(1, 3, 1), (0, 3, 1), (4, 2, 1), (3, 2, 1), (2, 2, 1), (1, 2, 1), (0, 2, 1), (4, 1, 1),(3, 1, 1), (2, 1, 1), (1, 1, 1), (0, 1, 1)},(4) h4,5(e0, e1, e2) = 0 or 1 for (e0, e1, e2) ∈ {(4, 3, 1), (3, 3, 1)} (still unknown).Remark 1.6. (1) n3(6, d) for 1 ≤ d ≤ 243 is not determined for 74 valuesof d (hence h6,3(e0, e1, e2, e3, e4) is unknown for the 74 cases), see [4].(2) It is known that h6,3(e0, 0, 1, 2, 2) = 2 for e0 = 0, 1, 2 since n3(6, d) = g3(6, d)+2 for d = 16, 17, 18 and since n3(5, 6) = g3(5, 6). Thus, hk,3(e0, e1, · · · , ek−2) ≥ 2could happen for k ≥ 6.(3) h3,q can be determined from the results on (n, r)-arcs in PG(2, q) since (n, n−d)-arcs and projective [n, 3, d]q codes are equivalent objects (recall that Griesmer[n, k, d]q codes with d ≤ qk−1 are projective), see [1]. For example, h3,q(0, 2) = 1holds for q ≥ 3 from the nonexistence of (q2− q− 1, q− 1)-arcs and the existenceof (q2 − q, q)-arcs in PG(2, q). But to find the largest n for which an (n, r)-arcexists in PG(2, q) for given r is a quite difficult problem in general, see [3].Proof of Theorem 1.1.Lemma 2.1. For an [n, k, d]q code, it holds that n ≥ gq(k, d) + t ifnq(k − 1, d′) = gq(k − 1, d′) + t for some integer t, where d′ = ⌈d/q⌉.Note on an Improvement of the Griesmer Bound for q-ary Linear Codes 203P r o o f. Let C be an [n, k, d]q code with d′ = ⌈d/q⌉, and let C′ be a residual[n− d, k − 1, d′]q code. From the assumption, we getn− d ≥ nq(k − 1, d′) = gq(k − 1, d′) + t.(2.1)Since d′ = ⌈d/q⌉ ≥ d/q, it holds that d′/qi ≥ d/qi+1, so we havegq(k − 1, d′) =k−2∑i=0⌈d′/qi⌉ ≥k−2∑i=0⌈d/qi+1⌉ =k−1∑i=1⌈d/qi⌉.Hence, from (2.1), we getn− d ≥k−1∑i=1⌈d/qi⌉+ t, i.e., n ≥ gq(k, d) + t. Remark 2.2. If d is an integer given by (1.1), thend′ = qk−2 −k−2∑i=1eiqi−1.(2.2)P r o o f o f T h e o r em 1.1. Since nq(k, d) ≥ gq(k, d), it is obvious from(1.3) that h3,q(e0, e1) ≥ 0. Hence (1.5) holds in the case andnq(3, d) = gq(3, d) + h3,q(e0, e1) for d = q2 − (e0 + e1q).In the case k = 4, nq(3, d′) ford = q3 − (e0 + e1q + e2q2), d′ = ⌈d/q⌉ = q2 − (e1 + e2q)(2.3)is expressed asnq(3, d′) = gq(3, d′) + h3,q(e1, e2).(2.4)Hence it follows from (2.3), (2.4) and Lemma 2.1 thatnq(4, d) ≥ gq(4, d) + h3,q(e1, e2), i.e., h4,q(e0, e1, e2) ≥ 0.In the case k ≥ 5, we shall prove (1.5) using induction on k. In this case, d andd′ can be expressed as (1.1) and (2.2), respectively. Sincenq(k − 1, d′) = gq(k − 1, d′) +k−1∑j=3hj,q(ek−j , ek−j+1, . . . , ek−2),204 Noboru Hamada, Tatsuya Marutait follows from Lemma 2.1 that the following inequality holds:nq(k, d) ≥ gq(k, d) +k−1∑j=3hj,q(ek−j , ek−j+1, . . . , ek−2),i.e., hk,q(e0, e1, . . . , ek−2) ≥ 0. Appendix. Tables of the values of d, e = e0e1 · · · ek−2, g = g3(k, d),n = n3(k, d) and hj = hj,3(ek−j , ek−j+1, · · · , ek−2) for 3 ≤ j ≤ k, for k = 3, 4, 5.Table 1. The values of g3(3, d), n3(3, d) and h3 for 1 ≤ d ≤ 9d e g n h31 22 3 3 02 12 4 4 03 02 5 6 14 21 7 7 05 11 8 8 06 01 9 9 07 20 11 11 08 10 12 12 09 00 13 13 0Table 2. The values of g3(4, d), n3(4, d) and h3, h4 for 1 ≤ d ≤ 27d e g n h3 h4 d e g n h3 h4 d e g n h3 h41 222 4 4 0 0 10 221 17 17 0 0 19 220 30 30 0 02 122 5 5 0 0 11 121 18 18 0 0 20 120 31 31 0 03 022 6 7 0 1 12 021 19 19 0 0 21 020 32 32 0 04 212 8 8 0 0 13 211 21 22 0 1 22 210 34 34 0 05 112 9 9 0 0 14 111 22 23 0 1 23 110 35 35 0 06 012 10 10 0 0 15 011 23 24 0 1 24 010 36 36 0 07 202 12 13 1 0 16 201 25 25 0 0 25 200 38 38 0 08 102 13 14 1 0 17 201 26 26 0 0 26 100 39 39 0 09 002 14 15 1 0 18 201 27 27 0 0 27 000 40 40 0 0Note on an Improvement of the Griesmer Bound for q-ary Linear Codes 205Table 3. The values of g3(5, d), n3(5, d) and h3, h4, h5 for 1 ≤ d ≤ 81d e g n h3 h4 h5 d e g n h3 h4 h51 2222 5 5 0 0 0 43 2011 66 67 0 1 02 1222 6 6 0 0 0 44 1011 67 68 0 1 03 0222 7 8 0 0 1 45 0011 68 69 0 1 04 2122 9 9 0 0 0 46 2201 71 72 0 0 15 1122 10 10 0 0 0 47 1201 72 73 0 0 16 0122 11 11 0 0 0 48 0201 73 74 0 0 17 2022 13 14 0 1 0 49 2101 75 76 0 0 18 1022 14 15 0 1 0 50 1101 76 77 0 0 19 0022 15 16 0 1 0 51 0101 77 78 0 0 110 2212 18 18 0 0 0 52 2001 79 79 0 0 011 1212 19 19 0 0 0 53 1001 80 80 0 0 012 0212 20 20 0 0 0 54 0001 81 81 0 0 013 2112 22 23 0 0 1 55 2220 85 85 0 0 014 1112 23 24 0 0 1 56 1220 86 86 0 0 015 0112 24 25 0 0 1 57 0220 87 87 0 0 016 2012 26 27 0 0 1 58 2120 89 89 0 0 017 1012 27 28 0 0 1 59 1120 90 90 0 0 018 0012 28 29 0 0 1 60 0120 91 91 0 0 019 2202 31 32 1 0 0 61 2020 93 94 0 0 120 1202 32 33 1 0 0 62 1020 94 95 0 0 121 0202 33 34 1 0 0 63 0020 95 96 0 0 122 2102 35 36 1 0 0 64 2210 98 98 0 0 023 1102 36 37 1 0 0 65 1210 99 99 0 0 024 0102 37 38 1 0 0 66 0210 100 100 0 0 025 2002 39 41 1 0 1 67 2110 102 102 0 0 026 1002 40 42 1 0 1 68 1110 103 103 0 0 027 0002 41 43 1 0 1 69 0110 104 104 0 0 028 2221 45 45 0 0 0 70 2010 106 106 0 0 029 1221 46 46 0 0 0 71 1010 107 107 0 0 030 0221 47 47 0 0 0 72 0010 108 108 0 0 031 2121 49 49 0 0 0 73 2200 111 111 0 0 032 1121 50 51 0 0 1 74 1200 112 112 0 0 033 0121 51 52 0 0 1 75 0200 113 113 0 0 034 2021 53 53 0 0 0 76 2100 115 115 0 0 035 1021 54 54 0 0 0 77 1100 116 116 0 0 036 0021 55 55 0 0 0 78 0100 117 117 0 0 037 2211 58 59 0 1 0 79 2000 119 119 0 0 038 1211 59 60 0 1 0 80 1000 120 120 0 0 039 0211 60 61 0 1 0 81 0000 121 121 0 0 040 2111 62 63 0 1 041 1111 63 64 0 1 042 0111 64 65 0 1 0206 Noboru Hamada, Tatsuya MarutaREFERENCES[1] Ball S. Table of bounds on three dimensional linear codes or (n, r) arcs inPG(2, q). http://www-ma4.upc.es/~simeon/codebounds.html[2] Griesmer J. H. A bound for error-correcting codes. IBM J. Res. Develop.,4 (1960), 532–542.[3] Hirschfeld J. W. P., L. Storme. The packing problem in statistics,coding theory and finite projective spaces. In: Finite Geometries, Proc. ofthe Fourth Isle of Thorns Conference, Kluwer, 2001(Eds A. Blokhuis et al.),201–246.[4] Maruta T. Griesmer bound for linear codes over finite fields.http://www.geocities.jp/mars39geo/griesmer.htm.[5] Solomon G., J. J. Stiffler.Algebraically punctured cyclic codes. Inform.Control, 8 (1965), 170–179.Noboru HamadaEmeritus Professor at Osaka Women’s Universitye-mail: n-hamada@koala.odn.ne.jpTatsuya MarutaDepartment of Mathematics and Information SciencesOsaka Prefecture UniversitySakai, Osaka 599-8531, Japane-mail: maruta@mi.s.osakafu-u.ac.jpReceived March 11, 2011Final Accepted June 2, 2011

Note on an Improvement of the Griesmer Bound for q-ary Linear Codes

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Note on an Improvement of the Griesmer Bound for q-ary Linear Codes

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