Polynomial support patterns simplify constacyclic code structure over rings
Support periods of polynomials and support-primitive decomposition of constacyclic codes over Galois rings
Information Theory
Summary
The paper studies certain types of error-correcting codes, called constacyclic codes, over complex algebraic structures known as Galois rings. It shows how breaking down these codes into simpler pieces based on patterns in polynomial coefficients helps understand their properties like error detection and correction strength. The authors connect these patterns to unique decompositions that reveal important distances and dual codes. Their work also provides efficient methods for calculations, especially when codes have simple constraints on their length.
What this means in practice
- •For coding engineers: Design constacyclic codes with guaranteed minimum distances and efficient factorization for error correction in communication systems over ring alphabets.
- •For cryptography developers: Use polynomial support and code decompositions to analyze and build secure ring-based cryptographic coding schemes.
Authors
Li Zhu, Hongfeng Wu
Abstract
Let \(A=\GR(p^e,r)\) have residue field \(\F_q\), and let \(λ\in A^\times\). We relate coefficient compression of monic divisors of \(X^N-λ\) to support-primitive decomposition of constacyclic codes. At a simple-root length, the support period of every nonconstant monic divisor is preserved at each coefficient precision and equals the order of the multiplicative stabilizer of its root set. For a cyclic Frobenius-orbit polynomial, the multiple equal-difference structure determines its support period, core degree, and least relative degree of an unramified extension over which all its basic-irreducible factors are binomials. Coordinate projections give every constacyclic code over a finite commutative ring a unique maximal interleaving into copies of one support-primitive code. At simple-root lengths over \(A\), the intrinsic period of a code is the greatest common divisor of the periods of its torsion codes. The decomposition yields distance and duality formulas and the arithmetic Singleton bound for free codes. At every length, a nonzero code over \(A\) has the same minimum Hamming distance as its highest torsion code, which gives a refined bound. At repeated-root lengths, identical torsion filtrations can occur with different intrinsic periods, while a class of nonzero codes obtained by substitution from simple-root generators retains the same core and minimum distance. Trace matrices recover the coefficients of basic-irreducible cores using unit pivots, and local Hensel corrections lift compressed factors directly. For \(X^n-1\) with \(p\nmid n\), lifting and output have quasi-linear bit cost in the compressed representation when the arithmetic data and residue factorizations are given.