Linear $$\ell $$-intersection pairs of codes and their applications

On pure MDS asymmetric entanglement-assisted quantum error-correcting codes

“…In this paper, we firstly complement the results in [14]. Specifically, we construct linear l-intersection pairs of two MDS codes with parameters [n, k 1 , n − k 1 + 1] q and [n, k 2 , n−k 2 +1] q , where (n, k 1 , k 2 , l) = (q, l+1, l+1, l) for 0 ≤ l ≤ q−2 and n = q + 1 with 1 ∈ {l, k 1 − l, k 2 − l} for k 1 , k 2 ≤ q.…”

“…Lemma 1 [14,Lemma 2.2] If there exists a linear l-intersection pair of two linear codes with parameters [n, k 1 ] q and [n,…”

“…In [14], Guenda et al gave some results about linear l-intersection pairs of MDS codes over F q with length n ≤ q + 1. In this section, we complement their results and give all possible linear l-intersection pairs of MDS codes over F 2 m with length n = 2 m + 2 ≥ 6.…”

“…First, let's recall some results in [14]. Guenda et al used the definition of the extended GRS codes as follows.…”

“…The required number of maximally entangled states of an AEAQECC is determined by the dimension of the intersection of the two corresponding linear codes. Two linear codes are said to be a linear l-intersection pair if their intersection has dimension l. In [14], Guenda et al constructed linear l-intersection pairs of MDS codes over F q with length up to q + 1 for most of the parameters.…”