Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number syndrome bits, and bit relies on some qubits. Constructing LDPC is challenging. It open problem to understand if there exist good i.e. with rate relative distance. Furthermore, techniques perform fault-tolerant gates poorly understood. ...

As a result of their applications in network coding, space-time coding, and coding for criss-cross errors, matrix codes have garnered significant attention; in various contexts, these codes have also been termed rank-metric codes, space-time codes over finite fields, and array codes. We focus on characterizing matrix codes that are both efficient (have high rate) and effective at error correcti...

A constant-composition code is a special constant-weight code under the restriction that each symbol should appear a given number of times in each codeword. In this correspondence, we give a lower bound for the maximum size of the -ary constant-composition codes with minimum distance at least 3. This bound is asymptotically optimal and generalizes the Graham–Sloane bound for binary constant-wei...

A t first Γ(L,G)-codes were introduced by V.D.Goppa [1] in 1970. These codes are a large and powerful class of error correcting codes. F.J. McWilliams and N.J. Sloane [2] defined these codes as the most important class of alternant codes. It is known that there are Γ(L,G)-codes that reach the Gilbert-Varshamov bound and that many Γ(L,G)-codes are placed in the Table of the best known codes [3]....

An $$(r,\ell )$$ -good polynomial is a of degree $$r+1$$ that constant on $$\ell $$ subsets $$\mathbb F_q$$ , each size . For any positive integer $$r\le 4$$ we provide an such =C_rq+O(\sqrt{q})$$ with $$C_r$$ maximal. This directly provides explicit estimate (up to error term $$O(\sqrt{q})$$ explict constant) for the maximal length and dimension Tamo–Barg LRC. Moreover, explain how construct g...

A spread is a set of lines of PG(d, q), which partition the point set. A parallelism is a partition of the set of lines by spreads. Some constructions of constant dimension codes that contain lifted MRD codes are based on parallelisms of projective spaces. It is known that there are 3 types of spreads in PG(3, 4) regular, subregular and aregular. A parallelism is regular if all its spreads are ...

An important code of length n is obtained by taking centralizer of a square matrix over a finite field Fq. Twisted centralizer codes, twisted by an element a ∈ Fq, are also similar type of codes but different in nature. The main results were embedded on dimension and minimum distance. In this paper, we have defined a new family of twisted centralizer codes namely generalized twisted centralizer...

As shown in [31], one of the five isomorphism types of optimal binary subspace codes of size 77 for packet length v = 6, constant dimension k = 3 and minimum subspace distance d = 4 can be constructed by first expurgating and then augmenting the corresponding lifted Gabidulin code in a fairly simple way. The method was refined in [36, 29] to yield an essentially computer-free construction of a ...