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Weighted projective spaces are natural generalizations of projective spaces with a rich structure. Projective Reed–Muller codes are error-correcting codes that played an important role in reliably transmitting information on digital communication channels. In this case study, we explore the power of commutative and homological algebraic techniques to study weighted projective Reed–Muller (WPRM) codes on weighted projective spaces of the form $\mathbb{P}(1,1,b)$. We compute minimal free resolutions and thereby obtain Hilbert series for the vanishing ideal of the ${𝔽}_q$-rational points, and compute main parameters for these codes.

We present a new approach to algorithm-based fault tolerance (ABFT) and parity-checking techniques in the design of high performance computing systems. The ABFT technique employs real convolution error-correcting codes to encode the input data. In order to reduce the round-off error from the output decoding process, systematic real convolution encoding is employed. This paper proposes an efficient method to detect the arithmetic errors using convolution codes at the output compared with an equivalent parity value derived from the input data. Number data processing errors are detected by comparing parity values associated with a convolution code. These comparable sets will be very close numerically, although not identical because of round-off error differences between the two parity generation processes. The effects of internal failures and round-off error are modeled by additive error sources located at the output of the processing block and input at threshold detector. This model combines the aggregate effects of errors and applies them to the respective outputs.

Any integral convex polytope $P$ in ${ℝ}^N$ provides an $N$-dimensional toric variety $X_P$ and an ample divisor $D_P$ on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on $X_P$, obtained by evaluating global section of the line bundle corresponding to $D_P$ on every rational point of $X_P$. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope $P$ and an algorithmic technique to get a lower bound on the minimum distance is described.

In this paper, we classify completely the negacyclic codes of length $3p^s$ over ${𝔽}_{p^m}+u{𝔽}_{p^m}+u^2{𝔽}_{p^m}$ when $u^3=0$ by using six-cyclotomic codes.

An MRD code ${𝒞}(n,k,n-k+1)$ is a $k$-dimensional ${𝔽}_{q^N}$-linear subspace of the $n$-dimensional vector space ${𝔽}_{q^N}^n$ over ${𝔽}_{q^N}$ for $n\le N$. Linear codes ${{𝒞}}^{\prime }(n^{\prime },k^{\prime },d)$ (of length $n^{\prime }\ge n$, dimension $k^{\prime }\ge k$) that are obtained from MRD codes ${𝒞}(n,k,d)$ satisfying the property ${{𝒞}}^{\prime }\cap {{{𝒞}}^{\prime }}^{\perp }=\{0\}$ are defined to be LCD MRD codes. An inherent relationship between the generator G and parity-check matrices H of LCD MRD codes is observed. This in fact identifies LCD MRD codes as trivial ($G{G}^T=I$) and nontrivial ($G{G}^T\ne I$) codes. Further, two classes of LCD MRD codes of length $n\ell$ are constructed from $\ell$ (trivial and nontrivial) LCD MRD codes of length $n$. Examples are provided for demonstration of results obtained.

In this paper, we use a method inspired by the valuation theory to classify completely cyclic codes of length $3p^s$ over ${𝔽}_{p^m}+u{𝔽}_{p^m}+u^2{𝔽}_{p^m}$. Then, we compute their separation parameters which was useful to compute the Hamming distance in [8].

The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG $\mathrm{\Gamma }$ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG $\mathrm{\Gamma }(v,k,t,\lambda ,\mu )$ with $\mu =1$, including some mixed Moore graphs and corresponding codes arising from them, in this paper.

Pseudorandomness is the theory of efficiently generating objects that "look random" despite being constructed with little or no randomness. One of the achievements of this research area has been the realization that a number of fundamental and widely studied "pseudorandom" objects are all almost equivalent when viewed appropriately. These objects include pseudorandom generators, expander graphs, list-decodable error-correcting codes, averaging samplers, and hardness amplifiers. In this survey, we describe the connections between all of these objects, showing how they can all be cast within a single "list-decoding framework" that brings out both their similarities and differences.