Jacobi polynomials and design theory II

2023/04/13 by Himadri Shekhar Chakraborty, Chakraborty, Himadri, Reina Ishikawa +3 · 1 citation
Computer Science · Engineering · #11F11 #Antenna Design and Optimization #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Primary 11T71 #Secondary 94B05 #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2304.06220

openalex publication_date 2023/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we introduce some new polynomials associated to linear codes over \mathbbF_q. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over \mathbbF_q, and give the MacWilliams type identity for it. We also give the notion of generalized q-colored t-designs. As an application of the generalized q-colored t-designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over \mathbbF_q.Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type~III and Type~IV codes.

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• Higher and extended Jacobi polynomials for codes

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