×

Local duality theorem for \(q\)-ary 1-perfect codes. (English) Zbl 1323.94155

Summary: In this paper, we derive the relationship between local weight enumerator of \(q\)-ary 1-perfect code in a face and that in the orthogonal face. As an application of our result, we compute the local weight enumerators of a shortened, doubly-shortened, and triply-shortened \(q\)-ary 1-perfect code.

MSC:

94B05 Linear codes (general theory)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Etzion T., Vardy A.: Perfect binary codes: constructions, properties, and enumeration. IEEE Trans. Inform. Theory 40(3), 754-763 (1994) · Zbl 0824.94029 · doi:10.1109/18.335887
[2] Hyun J.Y.: Generalized MacWilliams identities and their applications to perfect binary codes. Des. Codes Cryptogr. 50(2), 173-185 (2009) · Zbl 1237.94125 · doi:10.1007/s10623-008-9222-6
[3] Krotov D.S.: On weight distributions of perfect colorings and completely regular codes. Des. Codes Cryptogr. 61(3), 315-329 (2011) · Zbl 1235.94065 · doi:10.1007/s10623-010-9479-4
[4] MacWilliams F.J., Sloane N.J.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1998) · Zbl 0369.94008
[5] Vasil’eva A.Y.: Local spectra of perfect binary codes. Russian translations. II. Discret. Appl. Math. 135(1-3), 301-307 (2004) · Zbl 0930.94045 · doi:10.1016/S0166-218X(02)00313-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.