TY - JOUR
T1 - Variants of Jacobi polynomials in coding theory
AU - Chakraborty, Himadri Shekhar
AU - Miezaki, Tsuyoshi
N1 - Funding Information:
The authors thank Manabu Oura for helpful discussions. The authors would also like to thank the anonymous reviewers for their beneficial comments on an earlier version of the manuscript. The second named author is supported by JSPS KAKENHI (18K03217).
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2022/11
Y1 - 2022/11
N2 - In this paper, we introduce the notion of the complete joint Jacobi polynomial of two linear codes of length n over Fq and Zk. We give the MacWilliams type identity for the complete joint Jacobi polynomials of codes. We also introduce the concepts of the average Jacobi polynomial and the average complete joint Jacobi polynomial over Fq and Zk. We give a representation of the average of the complete joint Jacobi polynomials of two linear codes of length n over Fq and Zk in terms of the compositions of n and its distribution in the codes. Further we present a generalization of the representation for the average of the (g+ 1) -fold complete joint Jacobi polynomials of codes over Fq and Zk. Finally, we give the notion of the average Jacobi intersection number of two codes.
AB - In this paper, we introduce the notion of the complete joint Jacobi polynomial of two linear codes of length n over Fq and Zk. We give the MacWilliams type identity for the complete joint Jacobi polynomials of codes. We also introduce the concepts of the average Jacobi polynomial and the average complete joint Jacobi polynomial over Fq and Zk. We give a representation of the average of the complete joint Jacobi polynomials of two linear codes of length n over Fq and Zk in terms of the compositions of n and its distribution in the codes. Further we present a generalization of the representation for the average of the (g+ 1) -fold complete joint Jacobi polynomials of codes over Fq and Zk. Finally, we give the notion of the average Jacobi intersection number of two codes.
KW - Codes
KW - Jacobi polynomials
KW - Weight enumerators
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U2 - 10.1007/s10623-021-00923-2
DO - 10.1007/s10623-021-00923-2
M3 - Article
AN - SCOPUS:85113815334
SN - 0925-1022
VL - 90
SP - 2583
EP - 2597
JO - Designs, Codes, and Cryptography
JF - Designs, Codes, and Cryptography
IS - 11
ER -