Maximum distance separable (MDS) codes and near MDS (NMDS) codes play a fundamental role in coding theory due to their maximal error-correcting capabilities and rich algebraic structure. This paper constructs a new class of linear codes by twice extending TGRS codes, denoted as $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $. First, we establish the necessary and sufficient conditions for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ to be MDS. By analyzing the AMDS properties of both $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and their duals $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty)^{\bot} $, we determine the necessary and sufficient conditions for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ to be NMDS. Besides, we provide the form of the parity check matrix for $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and characterize the weight distributions of both the codes $ C_{k,n,\lambda_{1},\lambda_{2}, \delta}(\mathit{\boldsymbol{\alpha}},\mathit{\boldsymbol{v}},\infty) $ and their duals. Additionally, using the Schur method, the non-generalized Reed-Solomon (non-GRS) properties are proven. Finally, several classes of linear complementary dual (LCD) codes are constructed.

Let $ {\mathbb{F}}_{q^m} $ be the $ m $-th degree extension of the finite field $ {\mathbb{F}}_{q} $, for some prime power $ q $ and a positive integer $ m $. For $ n \in \mathbb{N} $, consider the rational function $ f(x) = \frac{f_{1}(x)}{f_{2}(x)} \in {\mathbb{F}}_{q^m}(x) $, where $ f_{1} $ and $ f_{2} $ are co-prime polynomials over $ {\mathbb{F}}_{q^m} $ with $ f_1 $ square-free, and deg($ f_{1} $)+deg($ f_{2} $$ ) = n $. In this article, we have established a condition which is sufficient to ensure the existence of a primitive pair $ (\alpha,f(\alpha))\in $ $ {\mathbb{F}}_{q^m}^*\times {\mathbb{F}}_{q^m}^* $ such that $ \mathrm{Qc}_{ {\mathbb{F}_{q^m}}/ {\mathbb{F}_{q}}}(\alpha) = a $, where $ a\in {\mathbb{F}}_{q}^* $ and $ \mathrm{Qc}_{ {\mathbb{F}_{q^m}}/ {\mathbb{F}_{q}}}(\alpha) = \underset{0\leq i<j\leq m-1}{\sum}\Bigg(\underset{k\neq i,j}{\underset{0\leq k\leq m-1}{\prod_{}^{}}}\alpha^{q^k}\Bigg) $. Moreover, when $ n = 2 $, $ m\geq 9 $ and $ q $ is an odd prime power, we show that such a pair will definitely exist, apart from $ 21 $ possible choices.

Let $ \mathbb{F}_q $ be the finite field with $ q $ elements where $ q = p^m $ for some prime $ p $ and $ m >0 $. In this article, we provide some constructions of Galois LCD codes over $ \mathbb{F}_q $ in terms of their generator matrices. Here, we consider the Galois inner product instead of Euclidean or Hermitian inner products. For these constructions, we use the matrix $ A $ for which $ A[\sigma^{m-l}(A)]^t = I,\; 0\leq l \leq m-1 $ where $ \sigma $ is the Frobenius map. We also define the Galois self-dual basis and use it to show the existence of a Euclidean LCD code from a linear code over some finite field. Further, we present an application of Galois LCD codes in a multi-secret sharing scheme. Besides, we obtain several optimal and almost optimal Galois LCD codes to show the importance of our results.

Driven by the applications in DNA storage, codes capable of correcting errors in DNA sequences have attracted considerable interest. This paper concentrates on a scenario in which sequences are affected by errors known as reverse-complement duplications. More precisely, we construct error-correcting codes for channel where a single reverse-complement duplication of arbitrary length may happen. In general, handling reverse-complement duplications of variable (non-fixed) lengths involves combinatorial challenges, and our construction is the first code capable of correcting such errors of arbitrary lengths. By investigating some special combinatorial structures of the errors, we propose an explicit code construction using duplication-free sequences. Furthermore, we show that as the alphabet size $ q $ varies as an increasing function of $ n $, the asymptotic rate is 1.

It is well known that both the minimum distances and stopping distances of LDPC codes are important measures of their performance. In this paper, we analyze the parameters of LDPC codes based on the totally isotropic subspaces of symplectic space and obtain the exact minimum distances and stopping distances of the codes under certain special circumstances. Additionally, we derive the upper and lower bounds on the dimensions of the resulting codes.

We study the hull, that is the intersection of a code with its orthogonal, of both linear and additive codes over the rings with unity of order $ 4, $ where the orthogonal is the Euclidean orthogonal for linear codes and for additive codes it is determined using characters. We relate the hull of the code with the hull of its image under the corresponding Gray map and use this to count the number of codes with a given hull for additive codes for the rings with characteristic $ 2 $. We investigate the codes over these rings which have a hull of given type which gives the cardinality of the hull.

In this article, we consider linear codes over the chain ring $ \frac{\mathbb{F}_{2}[x]}{\langle x^s\rangle} $ with $ s\geq 2 $ that are obtained using a construction referred to as the $ \mathcal{C}_D $-construction, where $ D $ is a set generated with the help of certain simplicial complexes. The main result is the utilization of the trace map $ \tau:\mathbb{F}_{2}[x]/\langle x^s\rangle\longrightarrow \mathbb{F}_2 $ given by $ a_0+a_1u+a_2u^2+\cdots+a_{s-1}u^{s-1}\mapsto a_0+a_1+\cdots+a_{s-1} $ to study binary subfield codes of $ \mathcal{C}_D $-codes. These binary subfield codes are $ l $-weight codes where $ l = 1, 2 $, or $ 4 $, and most of them are distance-optimal codes. Moreover, these binary codes are self-orthogonal and minimal under certain conditions. We present many new distance-optimal binary linear codes.

Linear codes achieving or nearly achieving the Singleton bound have many nice applications in data storage, combinatorial design, secret sharing, and so on. In this paper, we present several classes of MDS codes and NMDS codes, and the parameters of these codes are determined. In addition, by applying a result proposed by Tang and Ding (IEEE Trans. Inform. Theory, 67 (1), 244-254, 2021), we prove that some of these NMDS codes support $ t $-designs, where $ t = 2, 3 $. The parameters of these $ t $-designs are also derived.