Classification of perfect linear codes with crown poset structure

Brualdi et al. [Codes with a poset metric, Discrete Math. 147 (1995) 57-72] introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Golay code to be a 4-error-correcting perfect P-code. In this paper we classify all of the poset structures which admit the extended binary Golay code to be a 4-error-correcting perfect P-code, and show that there are no posets which admit the extended binary Golay code to be a 5-error-correcting perfect P-code.

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“…If the extended binary Golay code G 24 is a 4-error-correcting perfect P-code, then for any two distinct elements a, b in (1) …”

Section: Lemmamentioning

confidence: 99%

“…The theory over an arbitrary finite field can be treated in a similar manner. We refer to [1,2] for general theory.…”

Section: Introductionmentioning

confidence: 99%

The poset structures admitting the extended binary Golay code to be a perfect code

Jang

Kim

et al. 2008

Discrete Mathematics

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“…In [1,2,4] the authors defined an r -error-correcting perfect poset code as a code for which the r -balls centered at the codewords cover the whole space without overlapping. We modify this notion and introduce the notion of I -perfect poset codes, where I is an order ideal of P. Definition 2.8 Let P be a poset on [n] and I an order ideal of P. A linear P-code C over F q of length n is called I -perfect if the I -balls centered at the codewords of C are pairwise disjoint and their union is…”

Section: Proposition 27 Let P Be a Poset On [N]mentioning

confidence: 99%

Maximum distance separable poset codes

Hyun

Kim

2008

Des. Codes Cryptogr.

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We derive the Singleton bound for poset codes and define the MDS poset codes as linear codes which attain the Singleton bound. In this paper, we study the basic properties of MDS poset codes. First, we introduce the concept of I -perfect codes and describe the MDS poset codes in terms of I -perfect codes. Next, we study the weight distribution of an MDS poset code and show that the weight distribution of an MDS poset code is completely determined. Finally, we prove the duality theorem which states that a linear code C is an MDS P-code if and only if C ⊥ is an MDS P-code, where C ⊥ is the dual code of C and P is the dual poset of P.Keywords Maximum distance separable code · Poset code · Discrete Poisson summation formula · Moebius inversion formula AMS Classifications 94B05 · 94B25

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“…There are several papers [1], [3], [4] on the existence of 1-, 2-, or 3-error-correcting poset codes. The approach of the present work is opposite; we start to classify posets that admit the existence of perfect codes correcting as many as possible errors with respect to the code length and dimension, i. e., when the number of errors is close to the code codimension.…”

Section: Introductionmentioning

confidence: 99%