Coset weight enumerators and a Green’s function over a Hamming scheme with application to Cheeger’s constants
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We show that the Hamming weight enumerators of cosets of a linear code C over Fq of length n are closely related to a discrete Green’s function Gβ. Using this relation, for instance, we obtain the ex- act number of distinct coset weight enumerators of C. As an application, we show that lower bounds of the Cheeger ratio and the Cheeger con- stant of C on Γ₁ can be explicitly determined by Gβ and a coset weight enumerator value of C, where Γ₁is a distance regular graph with respect to a Hamming distance 1 over Fnq .
Abstract
We show that the Hamming weight enumerators of cosets of a linear code C over Fq of length n are closely related to a discrete Green’s function Gβ. Using this relation, for instance, we obtain the ex- act number of distinct coset weight enumerators of C. As an application, we show that lower bounds of the Cheeger ratio and the Cheeger con- stant of C on Γ₁ can be explicitly determined by Gβ and a coset weight enumerator value of C, where Γ₁is a distance regular graph with respect to a Hamming distance 1 over Fnq .
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