2025/08/16 by Chakraborty, Himadri Shekhar, Miezaki, Tsuyoshi
#11F11 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT) #Primary 11T71 #Secondary 94B05
paper · doi:10.48550/arxiv.2508.11909
In this paper, we introduce Jacobi polynomial generalizations of several classical invariants in coding theory over finite fields, specifically, the higher and extended weight enumerators, and we establish explicit correspondences between the resulting Jacobi polynomials. Moreover, we present the Jacobi analogue of MacWilliams identity for both higher and extended weight enumerators. We also present that the higher Jacobi polynomials for linear codes whose subcode supports form t-designs can be uniquely determined from the higher weight enumerators of the codes via polarization technique. Finally, we demonstrate how higher Jacobi polynomials can be computed from harmonic higher weight enumerators with the help of Hahn polynomials.