Cyclic codes length 7p power s over extension ring expanded and classified
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
This paper deals with a type of error-correcting code, which helps detect and fix errors in data. The authors study codes made from a special mathematical structure involving primes and powers, identifying how these codes break down into simpler parts. They classify these parts into four groups and exactly count them, providing explicit formulas for their properties. These findings help build new quantum error-correcting codes and codes with particular duality properties helpful in communication.
What this means in practice
- •For quantum communication engineers: Design quantum stabilizer codes with specific parameters derived from classified cyclic codes improving error correction in quantum systems.
- •For coding theory developers: Create linear codes with complementary duals tailored for secure and efficient communication channels using new parameter criteria from decomposed codes.
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).