[Paper Review] One generator $(1+u)$-quasi twisted codes over $F_2+uF_2$

Jian Gao, Qiong Kong|arXiv (Cornell University)|Jun 30, 2013

Coding theory and cryptography3 citations

TL;DR

This paper presents minimum generating sets for three types of one-generator (1+u)-quasi-twisted (QT) codes over the ring $F_2 + uF_2$ with $u^2 = 0$, and establishes a lower bound on the minimum Lee distance for a special class of $A_2$-type codes. Using the Gray map, these codes yield optimal or suboptimal linear codes over $F_2$, including examples like $[12,5,4]$, $[18,2,12]$, and $[24,3,12]$ codes.

ABSTRACT

This paper gives the minimum generating sets of three types of one generator $(1+u)$-quasi twisted (QT) codes over $F_2+uF_2$, $u^2=0$. Moreover, it discusses the generating sets and the lower bounds on the minimum Lee distance of a special class of $A_2$ type one generator $(1+u)$-QT codes. Some good (optimal or suboptimal) linear codes over $F_2$ are obtained by these types of one generator $(1+u)$-QT codes.

Motivation & Objective

• To determine the minimum generating sets for three types of one-generator (1+u)-quasi-twisted codes over the ring $F_2 + uF_2$.
• To analyze the structural properties of $A_2$-type (1+u)-QT codes and derive a lower bound on their minimum Lee distance.
• To construct good linear codes over $F_2$ using the Gray map from these (1+u)-QT codes.
• To provide explicit constructions and generating sets for $A_1$, $A_2$, and $B$-type codes via polynomial ideals and module homomorphisms.
• To demonstrate that the constructed codes achieve optimal or suboptimal parameters in $F_2$-linear codes via the Gray isometry.

Proposed method

• Uses the ring $R = F_2 + uF_2$ with $u^2 = 0$ and models codes as $R[x]$-submodules of $S_n^l = R[x]/(x^n - (1+u))^l$.
• Applies the isomorphism $\rho: R^N \to S_n^l$ to represent one-generator (1+u)-QT codes as $C = (f_0(x), \dots, f_{l-1}(x))$ with $f_j(x) \in S_n$.
• Employs module homomorphisms $\psi_i$ to project $C$ onto individual $(1+u)$-constacyclic codes in $S_n$, classifying $C$ as $A_1$, $A_2$, or $B$ type.
• Derives minimum generating sets using GCD and polynomial division: for $A_1$ type, $S_1 = \{G, xG, \dots, x^{r-1}G\}$ and $S_2 = \{F, xF, \dots, x^{n-r-1}F\}$ with $F = \{uf_0, \dots, uf_{l-1}\}$.
• For $A_2$-type codes, proves that the generating set is $\{G, xG, \dots, x^{r-1}G\}$ where $G = (ugf_0, \dots, ugf_{l-1})$, under the condition $\gcd(f_i, (x^n-1)/g) = 1$.
• Applies the Gray map $\phi$ to transform codes over $R$ into binary linear codes, preserving Lee distance to Hamming distance.

Experimental results

Research questions

• RQ1 What are the minimum generating sets for the three types of one-generator (1+u)-quasi-twisted codes over $F_2 + uF_2$?
• RQ2 How can the minimum Lee distance of $A_2$-type (1+u)-QT codes be bounded from below?
• RQ3 Can one-generator (1+u)-QT codes over $F_2 + uF_2$ yield optimal or suboptimal linear codes over $F_2$ via the Gray map?
• RQ4 What structural conditions ensure that the generating sets of $A_1$, $A_2$, and $B$-type codes are minimal and linearly independent?
• RQ5 How do the algebraic properties of the generating polynomials $g(x)$, $f_i(x)$, and $h(x)$ influence the code's size and minimum distance?

Key findings

• The minimum generating set for an $A_1$-type one-generator (1+u)-QT code is $S_1 \cup S_2$, where $S_1 = \{G, xG, \dots, x^{r-1}G\}$ and $S_2 = \{F, xF, \dots, x^{n-r-1}F\}$, with $r = \deg h$ and $h = (x^n - 1)/g$.
• For $A_2$-type codes, the minimum generating set is $\{G, xG, \dots, x^{r-1}G\}$, where $G = (ugf_0, \dots, ugf_{l-1})$, and $\gcd(f_i, (x^n-1)/g) = 1$ for all $i$.
• The minimum Lee distance of an $A_2$-type code satisfies $d_L(C) \geq l \cdot d_L(\widetilde{C})$, where $\widetilde{C} = (ug)$, and equality is achieved in examples.
• An $A_1$-type code with $n=3$, $l=2$, $g_0 = x+1$, $g_1 = x^2+1$ generates a $[12,5,4]$ linear code over $F_2$ via the Gray map, which is optimal.
• An $A_2$-type code with $n=3$, $l=3$, $g=x+1$, $f_0 = x^3+x+1$, $f_1 = x^3+x^2+1$, $f_2 = 1$ yields a $[18,2,12]$ code over $F_2$, which is optimal.
• An $B$-type code with $n=9$, $l=2$, $g=(x+1)(x^6+x^3+1)$, $q_0=x$, $q_1=x+x^2$ produces a $[36,10,8]$ linear code over $F_2$ via the Gray map.