Classification of special cyclic codes enables advances in quantum and LCD codes
Cyclic Codes of Length 7_p^s over F_p^m + uF_p^m : Characterization, Duals, and Applications to Quantum and LCD Codes
Information Theory
Summary
The paper studies a special kind of error-correcting code called cyclic codes, specifically those of a certain length over a mathematical structure involving two elements with a special relation. The authors find a way to break down these codes into simpler parts and categorize one part into four clear types. They also figure out the exact number of codes of each type and describe how to find codes that pair well or are dual to these. Using these results, they build new quantum error-correcting codes and identify helpful properties for codes used in data transmission that avoid certain kinds of overlap.
cyclic codesfinite fieldscyclotomic polynomialcode decompositiondual codesquantum stabilizer codesCSS constructionlinear complementary dual (LCD) codeserror-correcting codesgenerator polynomial
Authors
Payel Chandra, Kalyan Hansda
Abstract
Let $R_2 = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ ($u^2 = 0$), where $p$ is an odd prime and $m, s \in \mathbb{N}$. For $p \equiv 3, 5 \pmod 7$ with $\gcd(m, 6) = 1$, the cyclotomic polynomial $Φ_7(x)$ is irreducible over $\mathbb{F}_{p^m}$. This yields a direct sum decomposition $C = C_1 \oplus C_2$ for any cyclic code $C$ of length $7p^s$ over $R_2$, where $C_1$ has length $p^s$ and $C_2$ is a $7$-cyclotomic code of length $6p^s$. We classify $7$-cyclotomic codes into four disjoint generator-based types and calculate exact cardinalities using residue and torsion subcodes. Furthermore, explicit generators for the Euclidean dual codes $C^\perp$ are determined. As operational applications of these classified codes, we construct new families of quantum stabilizer codes via the CSS framework and establish parameter criteria for linear codes with complementary duals (LCD codes).