On the $\ell$-DLIPs of codes over finite commutative rings

TL;DR

This paper introduces the concept of $oldsymbol{ ext{ extit{ extl}}}$-dimension linear intersection pairs ($oldsymbol{ ext{ extl}}$-DLIPs) of codes over finite commutative rings, providing conditions for their existence and applications to quantum error correction.

Contribution

It generalizes existing duality concepts to $oldsymbol{ ext{ extl}}$-DLIPs over rings, offering a uniform method and new constructions for quantum codes.

Findings

Necessary and sufficient conditions for non-free and free $oldsymbol{ ext{ extl}}$-DLIPs.

Generator set for intersections of constacyclic codes over chain rings.

Construction of entanglement-assisted quantum error correcting codes from $oldsymbol{ ext{ extl}}$-DLIPs.

Abstract

Generalizing the linear complementary duals, the linear complementary pairs and the hull of codes, we introduce the concept of $\ell$-dimension linear intersection pairs ($\ell$-DLIPs) of codes over a finite commutative ring $(R)$, for some positive integer $\ell$. In this paper, we study $\ell$-DLIP of codes over $R$ in a very general setting by a uniform method. Besides, we provide a necessary and sufficient condition for the existence of a non-free (or free) $\ell$-DLIP of codes over a finite commutative Frobenius ring. In addition, we obtain a generator set of the intersection of two constacyclic codes over a finite chain ring, which helps us to get an important characterization of $\ell$-DLIP of constacyclic codes. Finally, the $\ell$-DLIP of constacyclic codes over a finite chain ring are used to construct new entanglement-assisted quantum error correcting (EAQEC) codes.