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The spectral relation between irreducible cyclic codes and generalized Paley graphs
Abstract
Let p be a prime and 𝔽q/𝔽r a finite field extension with q=pm and r=ps. For any k| q−1, we consider r-ary irreducible cyclic codes (ICC) of the form C(k,q/r) = {(Trq/r(γωik)i=0n−1)}γ∈ 𝔽r, with ω a primitive element of 𝔽q and $ n= \tfrac{q-1}{k}$, and generalized Paley (GP) graphs Γ(k,q) = Cay(𝔽q, { xk : x ∈ 𝔽q* }). We show that there is a simple closed formula relating the weight distribution of C(k,q/r) with the spectrum of Γ(kr,q), where kr=gcd(k, (q−1)/(r−1)). Then, we give $Spec(Γ(k,q))$ explicitly for those graphs associated with irreducible 2-weight cyclic codes in the semiprimitive and exceptional cases. Finally, we give the weight enumerators of irreducible cyclic codes associated with Hamming GP-graphs.
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BibTeX
@misc{podesta2026spectral,
title = {{The spectral relation between irreducible cyclic codes and generalized Paley graphs}},
author = {Ricardo A. Podestá and Denis E. Videla},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28546},
eprint = {2609.28546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28546},
}
RIS
TY - UNPB
TI - The spectral relation between irreducible cyclic codes and generalized Paley graphs
AU - Ricardo A. Podestá
AU - Denis E. Videla
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28546
UR - https://arxiv.org/abs/2609.28546
AB - Let p be a prime and 𝔽_q/𝔽_r a finite field extension with q=p^m and r=p^s. For any k| q−1, we consider r-ary irreducible cyclic codes (ICC) of the form C(k,q/r) = {(Tr_q/r(γω^ik)_i=0^n−1)}_γ∈ 𝔽_r, with ω a primitive element of 𝔽_q and $ n= \tfrac{q-1}{k}$, and generalized Paley (GP) graphs Γ(k,q) = Cay(𝔽_q, { x^k : x ∈ 𝔽_q^* }). We show that there is a simple closed formula relating the weight distribution of C(k,q/r) with the spectrum of Γ(k_r,q), where k_r=gcd(k, (q−1)/(r−1)). Then, we give $Spec(Γ(k,q))$ explicitly for those graphs associated with irreducible 2-weight cyclic codes in the semiprimitive and exceptional cases. Finally, we give the weight enumerators of irreducible cyclic codes associated with Hamming GP-graphs.
ER -