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We determine the minimum length n
q
(k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n
q
(k, d) = g
q
(k, d) + 1 for when k is odd, for when k is even, and for .
This work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD). (KRF-2005-214-C00175).
This research has been partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science
under Contract Number 17540129. 相似文献
4.
In this paper, we shall prove that there is no [3q4-q3-q2-3q-1,5,3q4-4q3-2q+1]q code over the finite field for q11. Thus, we conclude the nonexistence of a [gq(5,d),5,d]q code for 3q4-4q3-2q+1d3q4-4q3-q. 相似文献
5.
In this paper, we shall prove that the minimum length nq(5,d) is equal to gq(5,d) +1 for q4−2q2−2q+1≤ d≤ q4 − 2q2 − q and 2q4 − 2q3 − q2 − 2q+1 ≤ d ≤ 2q4−2q3−q2−q, where gq(5,d) means the Griesmer bound
.
Communicated by: J.D. Key 相似文献
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7.
Harold N. Ward 《Designs, Codes and Cryptography》2001,22(2):139-148
A divisibility theorem for codes meetingthe Griesmer bound is used to establish that there is no [207, 4, 165]code over GF(5). 相似文献
8.
《Discrete Mathematics》2022,345(12):113088
The quest to build large-scale quantum computing devices depends on keeping the noise level below a fault-tolerance threshold. In this paper we derive the asymmetric quantum analogue of the Griesmer bound. To benefit from the noise asymmetry in many physical systems, one can decide to only detect a single bit-flip error while maximizing control over the phase-flip errors. We present constructions of such codes via the classical Griesmer codes and obtain infinite families. The optimality of the parameters of the codes in the families is measured against the quantum Griesmer bound. Numerous other codes, which may not be optimal, can also be derived. Choices of their design provide greater flexibility in terms of the resulting quantum parameters. We give examples of good qubit, qutrit, and ququad codes from such a route. 相似文献
9.
Tatsuya Maruta 《Designs, Codes and Cryptography》2001,22(2):165-177
There do not exist
codes over the Galois field GF
attaining the Griesmer bound for
for
andfor
for
. 相似文献
10.
11.
Noboru Hamada 《Designs, Codes and Cryptography》1997,10(1):41-56
Let k and d be any integers such that k 4 and
. Then there exist two integers and in {0,1,2} such that
. The purpose of this paper is to prove that (1) in the case k 5 and (,) = (0,1), there exists a ternary
code meeting the Griesmer bound if and only if
and (2) in the case k 4 and (,) = (0,2) or (1,1), there is no ternary
code meeting the Griesmer bound for any integers k and d and (3) in the case k 5 and
, there is no projective ternary
code for any integers k and such that 1k-3, where
and
for any integer i 0. In the special case k=6, it follows from (1) that there is no ternary linear code with parameters [233,6,154] , [234,6,155] or [237,6,157] which are new results. 相似文献
12.
E. J. Cheon 《Designs, Codes and Cryptography》2009,51(1):9-20
In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g
q
(k, d) + 1, k, d]
q
code for sq
k-1 − sq
k-2 − q
s
− q
2 + 1 ≤ d ≤ sq
k-1 − sq
k-2 − q
s
with 3 ≤ s ≤ k − 2 and q ≥ s + 1. Then we get n
q
(k, d) = g
q
(k, d) + 1 for (k − 2)q
k-1 − (k − 1)q
k-2 − q
2 + 1 ≤ d ≤ (k − 2)q
k-1 − (k − 1)q
k-2, k ≥ 6, q ≥ 2k − 3; and sq
k-1 − sq
k-2 − q
s
− q + 1 ≤ d ≤ sq
k-1 − sq
k-2 − q
s
, s ≥ 2, k ≥ 2s + 1 and q ≥ 2s − 1.
This work was partially supported by the Com2MaC-SRC/ERC program of MOST/KOSEF (grant # R11-1999-054) and was partially supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-214-C00175). 相似文献
13.
Iliya G. Bouyukliev 《Discrete Mathematics》2009,309(12):4049-4317
In this work, we consider a classification of infinite families of linear codes which achieve the Griesmer bound, using the projective dual transform. We investigate the correspondence between families of linear codes with given properties via dual transform. 相似文献
14.
TATSUYA MARUTA 《Geometriae Dedicata》1997,65(3):299-304
Let
be the smallest integer n for which there exists a linear code of length n, dimension k and minimum Hamming distance d over the Galois field GF(q). In this paper we determine
for
for all q, using a geometric method. 相似文献
15.
Minihypers were introduced by Hamada to investigate linear codes meeting the Griesmer bound. Hamada (Bull Osaka Women’s Univ
24:1–47, 1985; Discrete Math 116:229–268, 1993) characterized the non-weighted minihypers having parameters , with k−1 > λ1 > λ2 > ... > λ
h
≥ 0, as the union of a λ1-dimensional space, λ2-dimensional space, ..., λ
h
-dimensional space, which all are pairwise disjoint. We present in this article a weighted version of this result. We prove
that a weighted -minihyper , with k−1 > λ1 > λ2 > ... > λ
h
≥ 0, is a sum of a λ1-dimensional space, λ2-dimensional space, ..., and λ
h
-dimensional space.
This research was supported by the Project Combined algorithmic and theoretical study of combinatorial structures between the Fund for Scientific Research Flanders-Belgium (FWO-Flanders) and the Bulgarian Academy of Sciences. This research
is also part of the FWO-Flanders project nr. G.0317.06 Linear codes and cryptography. 相似文献
16.
It is well known that the problem of determining the weight distributions of families of cyclic codes is, in general, notoriously difficult. An even harder problem is to find characterizations of families of cyclic codes in terms of their weight distributions. On the other hand, it is also well known that cyclic codes with few weights have a great practical importance in coding theory and cryptography. In particular, cyclic codes having three nonzero weights have been studied by several authors, however, most of these efforts focused on cyclic codes over a prime field. In this work we present a characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field. 相似文献
17.
Iwan M. Duursma 《Journal of Combinatorial Theory, Series A》2006,113(8):1677-1688
The van Lint-Wilson AB-method yields a short proof of the Roos bound for the minimum distance of a cyclic code. We use the AB-method to obtain a different bound for the weights of a linear code. In contrast to the Roos bound, the role of the codes A and B in our bound is symmetric. We use the bound to prove the actual minimum distance for a class of dual BCH codes of length q2−1 over Fq. We give cyclic codes [63,38,16] and [65,40,16] over F8 that are better than the known [63,38,15] and [65,40,15] codes. 相似文献
18.
We prove that there does not exist a [q4+q3−q2−3q−1, 5, q4−2q2−2q+1]q code over the finite field
for q≥ 5. Using this, we prove that there does not exist a [gq(5, d), 5, d]q code with q4 −2q2 −2q +1 ≤ d ≤ q4 −2q2 −q for q≥ 5, where gq(k,d) denotes the Griesmer bound.MSC 2000: 94B65, 94B05, 51E20, 05B25 相似文献