Characterizing equal weight polynomials to improve constacyclic MDS codes
Equal Hamming Weights at Fixed Root Order and Applications to Constacyclic MDS Codes
Information Theory
Summary
The paper studies when special polynomials that break down a certain big polynomial all share the same number of nonzero terms when their coefficients are written out. The authors provide a clear way to identify all such situations, showing that in most cases these polynomials have a predictable structure. They then use this insight to understand constacyclic MDS codes, a special kind of error-correcting code, proving new limitations and giving examples of when these codes have specific mathematical properties.
What this means in practice
- •For error correction engineers: Design constacyclic MDS codes with better guarantees on polynomial structure and code parameters by using the classification of equal weight irreducible polynomials.
- •For coding theorists: Assess feasibility and limitations in constructing constacyclic MDS codes by applying gcd and norm obstructions derived from polynomial weight uniformity.
Authors
Li Zhu, Hongfeng Wu
Abstract
For a fixed positive integer $n>1$, we study the prime powers $q$ coprime to $n$ for which all monic irreducible factors of $Φ_n$ over $\F_q$ have the same coefficient Hamming weight. Cyclotomic orbit coefficients give a finite, effective classification of these fields. Outside finitely many computable characteristics, every factor has the support of its characteristic-zero orbit polynomial; the uniform condition $p>2^{\varphi(n)}$ suffices. In each remaining characteristic, extension exponents reduce to divisors of $\ord_n(p)$, and an integer variance identity expressed by polynomial gcd degrees decides equal weight. The same coefficient data determine the full weight distribution. As an application, we prove that an MDS constacyclic code with an irreducible generator forces every irreducible polynomial of the same root order to have full support, including factors associated with different constacyclic constants. This yields a uniform gcd and norm obstruction. We develop its short-multiplier extension, give a prime-order cubic family, and establish an explicit quadratic-residue exclusion without a prime-square alphabet or a near-square length hypothesis.