On the uniqueness resp. Nonexistence of certain codes meeting the Griesmer bound

Tilborg, H.C.A. van

doi:10.1016/s0019-9958(80)90115-1

Cited by 52 publications

(18 citation statements)

References 5 publications

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“…In fact, some of optimal codes are given in [3]. Actually, codes meeting the GB have been extensively studied by Solomon et al [4], Belov [5] and Helleseth and van Tilborg [6][7][8][9][10]. It is found in [10,11] that a finite projective geometry can be used in constructing those codes.…”

Section: Introductionmentioning

confidence: 99%

Use of matroid theory to construct a class of good binary linear codes

Wang

Truong³

2014

IET Communications

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It is still an open challenge in coding theory how to design a systematic linear (n, k) − code C over GF(2) with maximal minimum distance d. In this study, based on matroid theory (MT), a limited class of good systematic binary linear codes (n, k, d) is constructed, where n = 2 k − 1 + • • • + 2 k − δ and d = 2 k − 2 + • • • + 2 k − δ − 1 for k ≥ 4, 1 ≤ δ < k. These codes are well known as special cases of codes constructed by Solomon and Stiffler (SS) back in 1960s. Furthermore, a new shortening method is presented. By shortening the optimal codes, we can design new kinds of good systematic binary linear codes with parameters n = 2 k − 1 + • • • + 2 k − δ − 3u and d = 2 k − 2 + • • • + 2 k − δ − 1 − 2u for 2 ≤ u ≤ 4, 2 ≤ δ < k. The advantage of MT over the original SS construction is that it has an advantage in yielding generator matrix on systematic form. In addition, the dual code C ⊥ with relative high rate and optimal minimum distance can be obtained easily in this study.

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Section: Introductionmentioning

confidence: 99%

Use of matroid theory to construct a class of good binary linear codes

Wang

Truong³

2014

IET Communications

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“…[35] constructed the first [24,5,12] subcode C~;/2 of g24, improving a previously known [24,5,8] subcode. Note that C~412 is unique [43], has only two non-zero weights 12 and 16, and has a [24,2,16] subcode C~416. As C~416 satisfies the Griesmer bound, it has a generator matrix of which each row has weight 16 [43], [18].…”

Section: These Are Denoted By A24(2d 12 ) B24(dlo + 2e7) C24(3d S )mentioning

confidence: 99%

“…Note that C~412 is unique [43], has only two non-zero weights 12 and 16, and has a [24,2,16] subcode C~416. As C~416 satisfies the Griesmer bound, it has a generator matrix of which each row has weight 16 [43], [18]. Hence it is easy to see that C~416 is unique.…”

Section: These Are Denoted By A24(2d 12 ) B24(dlo + 2e7) C24(3d S )mentioning

confidence: 99%

“…In particular, d( C16) = 3 which by shortening C 46 on a minimum weight codeword of Cir, using Lemma 1.2 implies the existence of a [43,7,20] code with non-zero weights 20,24,28. This is a contradiction to the classification of [43,7,20] due to Bouyuklieva and Jaffe [5]. Proof.…”

Section: A [Ao(c 46 ) a 20 (C 46 ) A 24 (C 46 ) A 28 (C 46 )]Tmentioning

confidence: 99%

“…Since q48 contains the all-one vector, the repetition code [48,1,48] must be the one dimensional subcode first appearing in the subcode chain. By [43] there is a unique [48,6,24] …”

Section: A [Ao(c 46 ) a 20 (C 46 ) A 24 (C 46 ) A 28 (C 46 )]Tmentioning

confidence: 99%

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