[Paper Review] Some Bounds on Binary LCD Codes

Lucky Galvez, Jon-Lark Kim+3 more|arXiv (Cornell University)|Jan 16, 2017

Coding theory and cryptography10 references4 citations

TL;DR

This paper establishes exact values of $LCD[n,k]$, the maximum minimum distance $d$ for binary $[n,k,d]$ LCD codes, for $1 \leq k \leq n \leq 12$, and proves that $LCD[n,n-i] = 2$ for $i \geq 2$ and $n \geq 2^i$. It also derives recursive upper bounds: $LCD[n,k] \leq LCD[n,k-1]$ for odd $k$, and $LCD[n,k] \leq LCD[n,k-2]$ for even $k$, using principal submatrices and generator matrix determinants.

ABSTRACT

A linear code with a complementary dual (or LCD code) is defined to be a linear code $C$ whose dual code $C^{\perp}$ satisfies $C \cap C^{\perp}$= $\left\{ \mathbf{0} ight\} $. Let $LCD{[}n,k{]}$ denote the maximum of possible values of $d$ among $[n,k,d]$ binary LCD codes. We give exact values of $LCD{[}n,k{]}$ for $1 \le k \le n \le 12$. We also show that $LCD[n,n-i]=2$ for any $i\geq2$ and $n\geq2^{i}$. Furthermore, we show that $LCD[n,k]\leq LCD[n,k-1]$ for $k$ odd and $LCD[n,k]\leq LCD[n,k-2]$ for $k$ even.

Motivation & Objective

• To determine exact values of $LCD[n,k]$, the maximum minimum distance $d$ for binary $[n,k,d]$ LCD codes, for small parameters $n \leq 12$.
• To establish structural bounds on $LCD[n,k]$ based on code dimension $k$, showing it is non-increasing in $k$ under specific parity conditions.
• To investigate the existence and non-existence of LCD codes with high minimum distance, particularly for $n$ even or odd and $d$ close to $n$.
• To correct and update previously reported values of $LCK[n,d]$ (maximum $k$ for given $n,d$) using new bounds and exhaustive search.

Proposed method

• The authors use the Griesmer bound to derive upper limits on $d$ for $[n,k,d]$ codes, particularly for $k=2$, and construct explicit generator matrices to achieve these bounds.
• They apply Proposition 2.2, which states that a code is LCD if and only if $\det(GG^T) \neq 0$, to verify LCD properties of constructed codes.
• For recursive bounds, they analyze principal submatrices of generator matrices and use rank arguments to relate $LCD[n,k]$ to $LCD[n,k-1]$ and $LCD[n,k-2]$.
• They prove that $LCD[n,n-i] = 2$ for $i \geq 2$ and $n \geq 2^i$ by showing that no such code can have $d > 2$ due to parity and dimension constraints.
• They use MAGMA-based exhaustive search to compute $LCD[n,k]$ for small $n,k$, validating results and correcting prior errors in $LCK[n,d]$ tables.
• They derive general non-existence conditions: $LCK[n,n-2i] = 0$ for even $n > 6i$, and $LCK[n,n-2i-1] = 0$ for odd $n > 6i+3$, using the Griesmer bound and parity arguments.

Experimental results

Research questions

• RQ1 What are the exact values of $LCD[n,k]$ for all $1 \leq k \leq n \leq 12$?
• RQ2 Can the maximum minimum distance $d$ for binary LCD codes be bounded recursively in terms of $k$, and how does parity of $k$ affect this?
• RQ3 For $n \geq 2^i$ and $i \geq 2$, is $LCD[n,n-i]$ always equal to 2, and why?
• RQ4 Why do certain high-distance LCD codes (e.g., with $d=n$ or $d=n-1$) not exist for even $n$?
• RQ5 How can previously reported $LCK[n,d]$ values be corrected using new theoretical bounds and exhaustive computation?

Key findings

• Exact values of $LCD[n,k]$ are computed for all $1 \leq k \leq n \leq 12$, with $LCD[n,2] = \left\lfloor \frac{2n}{3} \right\rfloor$ for $n \equiv 1, \pm 2, 3 \pmod{6}$.
• For $i \geq 2$ and $n \geq 2^i$, it is proven that $LCD[n,n-i] = 2$, showing a sharp upper bound on minimum distance for codes of dimension $n-i$.
• $LCD[n,k]$ is non-increasing in $k$: $LCD[n,k] \leq LCD[n,k-1]$ for odd $k$, and $LCD[n,k] \leq LCD[n,k-2]$ for even $k$, established via principal submatrix rank analysis.
• The paper corrects erroneous values in prior literature for $LCK[n,d]$, particularly identifying $LCK[n,d] = 0$ for $n$ even and $d = n-2i$ when $n > 6i$, and for $n$ odd and $d = n-2i-1$ when $n > 6i+3$.
• An exhaustive MAGMA search confirms the computed $LCD[n,k]$ values and validates the theoretical bounds for small parameters.
• The maximum possible minimum distance for $[n,k,d]$ binary LCD codes is shown to be constrained by the Griesmer bound and parity, especially when $k=2$.