Linear Codes Of 2-Designs As Subcodes Of The Extended Generalized Reed-Muller Codes
Zhiwen He, Jiejing Wen
TL;DR
This paper investigates affine-invariant ternary codes derived from Hermitian functions, showing they form 2-designs supported by minimum weight codewords and are subcodes of extended generalized Reed-Muller codes, with detailed parameters provided.
Contribution
It establishes a connection between affine-invariant ternary codes and 2-designs, identifying these codes as subcodes of extended generalized Reed-Muller codes and analyzing their parameters.
Findings
Incidence matrices support 2-designs
Codes are subcodes of extended Reed-Muller codes
Dimensions and minimum weight bounds are provided
Abstract
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear codes generated by the rows of these incidence matrix are subcodes of the extended codes of the 4-th order generalized Reed-Muller codes and they also hold 2-designs. Finally, we give the dimensions and lower bound of the minimum weights of these linear codes.
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Finite Group Theory Research