Related papers: On the Decoding Complexity of Cyclic Codes Up to t…
The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root…
This work introduces a decoding strategy for binary self-dual codes possessing an automorphism of a specific type. The proposed algorithm is a hard decision iterative decoding scheme. The enclosed experiments show that the new decoding…
The hulls of linear and cyclic codes over finite fields have been of interest and extensively studied due to their wide applications. In this paper, the hulls of cyclic codes of length $n$ over the ring $\mathbb{Z}_4$ have been focused on.…
Staircase codes (SCCs) are typically decoded using iterative bounded-distance decoding (BDD) and hard decisions. In this paper, a novel decoding algorithm is proposed, which partially uses soft information from the channel. The proposed…
Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, have a special place in algebraic coding theory. Among other things, many of the best-known or optimal codes have been obtained from these classes. In this…
We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in…
There are many results on the minimum distance of a cyclic code of the form that if a certain set T is a subset of the defining set of the code, then the minimum distance of the code is greater than some integer t. This includes the BCH,…
We present two sequences of ensembles of non-systematic irregular repeat-accumulate codes which asymptotically (as their block length tends to infinity) achieve capacity on the binary erasure channel (BEC) with bounded complexity per…
Machine learning has the potential to become an important tool in quantum error correction as it allows the decoder to adapt to the error distribution of a quantum chip. An additional motivation for using neural networks is the fact that…
Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic…
For a class $\mathcal{C}$ of binary linear codes, we write $\theta_{\mathcal{C}}\colon (0,1) \to [0,\frac{1}{2}]$ for the maximum-likelihood decoding threshold function of $\mathcal{C}$, the function whose value at $R \in (0,1)$ is the…
his study presents a novel technique to estimate the computational complexity of sequential decoding using the Berry-Esseen theorem. Unlike the theoretical bounds determined by the conventional central limit theorem argument, which often…
It is shown that the counting function of n Boolean variables can be implemented with the formulae of size O(n^3.06) over the basis of all 2-input Boolean functions and of size O(n^4.54) over the standard basis. The same bounds follow for…
All utility-scale quantum computers will require some form of Quantum Error Correction in which logical qubits are encoded in a larger number of physical qubits. One promising encoding is known as the colour code which has broad…
Generalized-bicycle (GB) and more general two-block group-algebra (2BGA) quantum error-correcting codes have naturally redundant minimum-weight stabilizer generators. To use this redundancy, we constructed a large number of ``planar'' 2BGA…
We demonstrate a majority-logic decoding algorithm for decoding the generalised hyperoctahedral group $C_m \wr S_n$ when thought of as an error-correcting code. We also find the complexity of this decoding algorithm and compare it with that…
Quantum computers require error correction to achieve universal quantum computing. However, current decoding of quantum error-correcting codes relies on classical computation, which is slower than quantum operations in superconducting…
We present new upper and lower bounds on the minimum distance of certain generalized bicycle (GB) codes beyond the reach of techniques for classical codes capable of even capturing the true minimum distance for some cases. These bounds are…
This paper revisits the ordered statistics decoding (OSD). It provides a comprehensive analysis of the OSD algorithm by characterizing the statistical properties, evolution and the distribution of the Hamming distance and weighted Hamming…
The recently introduced polar codes constitute a breakthrough in coding theory due to their capacityachieving property. This goes hand in hand with a quasilinear construction, encoding, and successive cancellation list decoding procedures…