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The nonexistence of near-extremal formally self-dual codes

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Abstract

A code \({\mathcal {C}}\) is called formally self-dual if \({\mathcal {C}}\) and \({\mathcal {C}^{\perp}}\) have the same weight enumerators. There are four types of nontrivial divisible formally self-dual codes over \({\mathbb {F}_2,\,\mathbb {F}_3}\) , and \({\mathbb F_4}\) . These codes are called extremal if their minimum distances achieve the Mallows-Sloane bound. S. Zhang gave possible lengths for which extremal self-dual codes do not exist. In this paper, we define near-extremal formally self-dual (f.s.d.) codes. With Zhang’s systematic approach, we determine possible lengths for which the four types of near-extremal formally self-dual codes as well as the two types of near-extremal formally self-dual additive codes cannot exist. In particular, our result on the nonexistence of near-extremal binary f.s.d. even codes of any even length n completes all the cases since only the case 8|n was dealt with by Han and Lee.

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Correspondence to Sunghyu Han.

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Communicated by J.D. Key.

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Han, S., Kim, JL. The nonexistence of near-extremal formally self-dual codes. Des. Codes Cryptogr. 51, 69–77 (2009). https://doi.org/10.1007/s10623-008-9244-0

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  • DOI: https://doi.org/10.1007/s10623-008-9244-0

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