Channel coding plays a pivotal role in ensuring reliable communication over wireless channels. With the growing need for ultra-reliable communication in emerging wireless use cases, the significance ...of channel coding has amplified. Furthermore, minimizing decoding latency is crucial for critical-mission applications, while optimizing energy efficiency is paramount for mobile and the Internet of Things (IoT) communications. As the fifth generation (5G) of mobile communications is currently in operation and 5G-advanced is on the horizon, the objective of this paper is to assess prominent channel coding schemes in the context of recent advancements and the anticipated requirements for the sixth generation (6G). In this paper, after considering the potential impact of channel coding on key performance indicators (KPIs) of wireless networks, we review the evolution of mobile communication standards and the organizations involved in the standardization, from the first generation (1G) to the current 5G, highlighting the technologies integral to achieving targeted KPIs such as reliability, data rate, latency, energy efficiency, spectral efficiency, connection density, and traffic capacity. Following this, we delve into the anticipated requirements for potential use cases in 6G. The subsequent sections of the paper focus on a comprehensive review of three primary coding schemes utilized in past generations and their recent advancements: low-density parity-check (LDPC) codes, turbo codes (including convolutional codes), and polar codes (alongside Reed-Muller codes). Additionally, we examine alternative coding schemes like Fountain codes (also known as rate-less codes), sparse regression codes, among others. Our evaluation includes a comparative analysis of error correction performance and the performance of hardware implementation for these coding schemes, providing insights into their potential and suitability for the upcoming 6G era. Lastly, we will briefly explore considerations such as higher-order modulations and waveform design, examining their contributions to enhancing key performance indicators in conjunction with channel coding schemes.
We classify all <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>-ary <inline-formula> <tex-math notation="LaTeX">\Delta </tex-math></inline-formula>-divisible linear codes ...which are spanned by codewords of weight <inline-formula> <tex-math notation="LaTeX">\Delta </tex-math></inline-formula>. The basic building blocks are the simplex codes, and for <inline-formula> <tex-math notation="LaTeX">q=2 </tex-math></inline-formula> additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight 4 have been classified, which is the case <inline-formula> <tex-math notation="LaTeX">q=2 </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\Delta =4 </tex-math></inline-formula> of our classification. As an application, we give an alternative proof of a theorem of Liu on binary <inline-formula> <tex-math notation="LaTeX">\Delta </tex-math></inline-formula>-divisible codes of length <inline-formula> <tex-math notation="LaTeX">4\Delta </tex-math></inline-formula> in the projective case.
We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise ...structure of the codes, our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance. An important consequence of this result is that a sequence of Reed-Muller codes with increasing block length and converging rate achieves capacity. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to all affine-invariant codes and, thus, to extended primitive narrow-sense BCH codes. This also resolves, in the affirmative, the existence question for capacity-achieving sequences of binary cyclic codes. The primary tools used in the proof are the sharp threshold property for symmetric monotone Boolean functions and the area theorem for extrinsic information transfer functions.
Integrated Interleaved (II) and Extended Integrated Interleaved (EII) codes are a versatile alternative for Locally Recoverable (LRC) codes, since they require fields of relatively small size. II and ...EII codes are generally defined over Reed-Solomon type of codes. A new comprehensive definition of EII codes is presented, allowing for EII codes over any field, and in particular, over the binary field GF(2) . The traditional definition of II and EII codes is shown to be a special case of the new definition. Improvements over previous constructions of LRC codes, in particular, for binary codes, are given, as well as cases meeting an upper bound on the minimum distance. Properties of the codes are presented as well, in particular, an iterative decoding algorithm on rows and columns generalizing the iterative decoding algorithm of product codes. Two applications are also discussed: one is finding a systematic encoding of EII codes such that the parity symbols have a balanced distribution on rows, and the other is the problem of ordering the symbols of an EII code such that the maximum length of a correctable burst is achieved.
A Goppa code over <inline-formula> <tex-math notation="LaTeX">\Bbb F_{q^{m}} </tex-math></inline-formula> is a well-known subclass of algebraic error-correcting code. If <inline-formula> <tex-math ...notation="LaTeX">m=1 </tex-math></inline-formula>, then it is a generalized Reed-Solomon(GRS) code and its dual code is called a GRS code via a Goppa code. In this paper, we give a necessary and sufficient condition that the dual codes of GRS codes via (expurgated) Goppa codes are also GRS codes via Goppa codes. Under the above condition, we show that the hulls of GRS codes via Goppa codes are still GRS codes via Goppa codes. As an application, we characterize LCD GRS codes and self-dual GRS codes under the above condition. Some numerical examples are also presented to illustrate our main results. Moreover, we also apply our result to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs with arbitrary parameters.
Regenerating codes and codes with locality are two coding schemes that have recently been proposed, which in addition to ensuring data collection and reliability, also enable efficient node repair. ...In a situation where one is attempting to repair a failed node, regenerating codes seek to minimize the amount of data downloaded for node repair, while codes with locality attempt to minimize the number of helper nodes accessed. This paper presents results in two directions. In one, this paper extends the notion of codes with locality so as to permit local recovery of an erased code symbol even in the presence of multiple erasures, by employing local codes having minimum distance >2. An upper bound on the minimum distance of such codes is presented and codes that are optimal with respect to this bound are constructed. The second direction seeks to build codes that combine the advantages of both codes with locality as well as regenerating codes. These codes, termed here as codes with local regeneration, are codes with locality over a vector alphabet, in which the local codes themselves are regenerating codes. We derive an upper bound on the minimum distance of vector-alphabet codes with locality for the case when their constituent local codes have a certain uniform rank accumulation property. This property is possessed by both minimum storage regeneration (MSR) and minimum bandwidth regeneration (MBR) codes. We provide several constructions of codes with local regeneration which achieve this bound, where the local codes are either MSR or MBR codes. Also included in this paper, is an upper bound on the minimum distance of a general vector code with locality as well as the performance comparison of various code constructions of fixed block length and minimum distance.
The binary quadratic-residue codes and the punctured Reed-Muller codes <inline-formula> <tex-math notation="LaTeX">{\mathcal {R}}_{2}((m-1)/2, m)) </tex-math></inline-formula> are two families of ...binary cyclic codes with parameters <inline-formula> <tex-math notation="LaTeX">n, (n+1)/2, d \geq \sqrt {n} </tex-math></inline-formula>. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length <inline-formula> <tex-math notation="LaTeX">2^{m}-1 </tex-math></inline-formula> and dimension near <inline-formula> <tex-math notation="LaTeX">2^{m-1} </tex-math></inline-formula> with good minimum distances. When <inline-formula> <tex-math notation="LaTeX">m \geq 3 </tex-math></inline-formula> is odd, the codes become a family of duadic codes with parameters <inline-formula> <tex-math notation="LaTeX">2^{m}-1, 2^{m-1}, d </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">d \geq 2^{(m-1)/2}+1 </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">m \equiv 3 \pmod {4} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">d \geq 2^{(m-1)/2}+3 </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">m \equiv 1 \pmod {4} </tex-math></inline-formula>. The two families of binary cyclic codes contain some optimal binary cyclic codes.
In this paper, we construct protograph-based spatially coupled low-density parity-check (LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single ...coupled chain. By varying L , we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L . We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L , the BP thresholds on both the binary erasure channel and the binary-input additive white Gaussian noise channel saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a posteriori decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output channels with low-complexity BP decoding.
Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code optimal if it achieves the Singleton-type bound). In the breakthrough work ...of Tamo and Barg, several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given by Tamo and Barg are upper bounded by the code alphabet size q. Recently, it was proved through the extension of construction by Tamo and Barg that the length of q-ary optimal locally repairable codes can be q +1 by Jin et al. Surprisingly, Barg et al. presented a few examples of q-ary optimal locally repairable codes of small distance and locality with code length achieving roughly q 2 . Very recently, it was further shown in the work of Li et al. that there exist q-ary optimal locally repairable codes with the length bigger than q+1 and the distance proportional to n. Thus, it becomes an interesting and challenging problem to construct new families of q-ary optimal locally repairable codes of length bigger than q+1. In this paper, we construct a class of optimal locally repairable codes of distances 3 and 4 with unbounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen.
The hull of linear codes has promising utilization in coding theory and quantum coding theory. In this paper, we study the hull of generalized Reed-Solomon codes and extended generalized Reed-Solomon ...codes over finite fields with respect to the Euclidean inner product. Several infinite families of MDS codes with hulls of arbitrary dimensions are presented. As an application, using these MDS codes with hulls of arbitrary dimensions, we construct several new infinite families of entanglement-assisted quantum error-correcting codes with flexible parameters.
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