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A RECURSIVE FORMULA FOR WEIGHT DISTRIBUTION OF HAMMING CODES

D. S. Kim
μž₯μ „μˆ˜ν•™νšŒ (Jangjeon Mathematical Society)|2024.10|Proceedings of the Jangjeon Mathematical Society(μž₯μ „μˆ˜ν•™νšŒ 논문집) (Proceedings of the Jangjeon Mathematical Society)|제27ꢌ 제4호|pp. 871-879|DOI 10.17777/pjms.2024.27.4.030

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쑰회

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Hamming codeweight distributionPless power moment identityexponential sum
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초둝

We derive a recursive formula determining the weight distribution of the 𝑛 = ( π‘ž π‘š βˆ’ 1 ) / ( π‘ž βˆ’ 1 ) , 𝑛 βˆ’ π‘š , 3 n=(q m βˆ’1)/(qβˆ’1),nβˆ’m,3 Hamming code H(m, q). Here q is a prime power. The proof is based on Moisio’s idea of using Pless power moment identity together with exponential sum techniques. We first prove the recursive formula under the restriction (m, q βˆ’ 1) = 1. Then we remove the restriction so that the formula actually holds for any positive integer m and any prime power q.

Abstract

We derive a recursive formula determining the weight distribution of the 𝑛 = ( π‘ž π‘š βˆ’ 1 ) / ( π‘ž βˆ’ 1 ) , 𝑛 βˆ’ π‘š , 3 n=(q m βˆ’1)/(qβˆ’1),nβˆ’m,3 Hamming code H(m, q). Here q is a prime power. The proof is based on Moisio’s idea of using Pless power moment identity together with exponential sum techniques. We first prove the recursive formula under the restriction (m, q βˆ’ 1) = 1. Then we remove the restriction so that the formula actually holds for any positive integer m and any prime power q.

μ°Έκ³ λ¬Έν—Œ (11)

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  2. W. C. Huffman, "Fundamentals of Error-Correcting Codes" (2003), Cambridge University Press
  5. F. J. MacWilliams, "The Theory of Error Correcting Codes" (1998)
  6. M. Moisio, "The moments of a Kloosterman sum and the weight distribution of a Zetterberg type binary cyclic code", IEEE Trans. Inf. Theory, 53, pp.843~847 (2007)
  7. M. Moisio, "On the moments of Kloosterman sums and fibre products of Kloosterman curves", Finite Fields Appl, 14(2), pp.515~531 (2008)
  8. M. Moisio, "On the number of rational points on some families of Fermat curves over finite fields", Finite Fields Appl, 13, pp.546~562 (2007)
  9. V. S. Pless, "Handbook of Coding Theroy vol. I", pp.3~139 (1998)
  10. R. Schoof, "Hecke operators and the weight distribution of certain codes", J. Combin. Theroy Ser. A, 57, pp.163~186 (1991)
  11. G. van der Geer, "Weight formulas for ternary Melas codes", Math. Comp, 58, pp.781~792 (1992)

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