RM Codes Research Articles

Pipelined Architecture for Soft-Decision Iterative Projection Aggregation Decoding for RM Codes

Pipelined Architecture for Soft-Decision Iterative Projection Aggregation Decoding for RM Codes

Neural decoders with permutation invariant structure

Neural decoders were introduced as a generalization of the classic Belief Propagation (BP) decoding algorithms. In this work, we propose several neural decoders with different permutation invariant structures for BCH codes and punctured RM codes. Firstly, we propose the cyclically equivariant neural decoder which makes use of the cyclically invariant structure of these two codes. Next, we propose an affine equivariant neural decoder utilizing the affine invariant structure of those two codes. Both these two decoders outperform previous neural decoders when decoding cyclic codes. The affine decoder achieves a smaller decoding error probability than the cyclic decoder, but it usually requires a longer running time. Similar to using the property of the affine invariant property of extended BCH codes and RM codes, we propose the list decoding version of the cyclic decoder that can significantly reduce the frame error rate(FER) for these two codes. For certain high-rate codes, the gap between the list decoder and the Maximum Likelihood decoder is less than 0.1 dB when measured by FER.

Reed–Muller Network Coded-Cooperation With Joint Decoding

This letter proposes a Reed–Muller network coded-cooperative (RMNCC) scheme with the joint decoding. The Kronecker product (KP) construction is utilized to construct RM codes. The KP construction is effectively employed to accomplish network coded-cooperation between two users. An asymptotic error performance bound is presented to validate the average BER performance of the RMNCC scheme over fast Rayleigh fading channel. Monte Carlo simulated results for the RMNCC scheme agree with the analytical framework at higher SNR. Furthermore, a joint bit selection algorithm for the optimum selection of information bits at the relay node is also proposed, which results in an improved BER performance.

Spherically Punctured Reed-Muller Codes

Consider a binary Reed–Muller code RM $(r,m)$ defined on the $m$ -dimensional hypercube $\mathbb {F}_{2}^{m}$ . In this paper, we study punctured Reed–Muller codes $P_{r}(m,b)$ , whose positions are restricted to the $m$ -tuples of a given Hamming weight $b$ . In combinatorial terms, this paper concerns $m$ -variate Boolean polynomials of any degree $r$ , which are evaluated on a Hamming sphere of some radius $b$ in $\mathbb {F}_{2}^{m}$ . Codes $P_{r}(m,b)$ inherit some recursive properties of RM codes. In particular, they can be built from the shorter codes, by decomposing a spherical $b$ -layer into sub-layers of smaller dimensions. However, these sub-layers have different sizes and do not form the classical Plotkin construction. We analyze recursive properties of the spherically punctured codes $P_{r}(m,b)$ and find their distances for the arbitrary values of parameters $r,m$ , and $b$ . Finally, we describe recursive (successive cancellation) decoding of these codes.

Mixed polarized constructions

We consider Plotkin-type constructions that perform a multi-step recursive decomposition of a given code into the shorter codes and are similar to polar and Reed-Muller RM codes. However, we end this decomposition process at the various short codes instead of the single information bits used as end nodes in polar design. We also employ maximum likelihood ML decoding of the end codes. Such a design can reduce the output error rates of polarised constructions on the moderate blocklengths. We also analyse the complexity-performance trade-offs in order to optimise code design.

Integrating CE and Modified SLM to Reduce the PAPR of OFDM Systems

This study primarily integrated the techniques of selected mapping (SLM), constellation extension (CE), and Reed---Muller codes (RM codes) to construct an improved SLM technique termed RM-CE-SLM. A high peak-to-average power ratio (PAPR) is the primary disadvantage of orthogonal frequency division multiplexing (OFDM) systems. High PAPR not only reduces the performance of high-power amplifiers, but also increases the complexity of digital-to-analog converters. The RM-CE-SLM technique is primarily used to improve the high PAPR phenomenon of OFDM systems. The RM-CE-SLM technique initially uses RM codes to select the internal and external positions of constellation points in a 16 quadrature amplitude modulation (16-QAM) CE diagram. The technique also uses codeword elements in cyclic codes to produce the SLM scrambling sequence. The elements of a scrambling sequence are based on the codeword elements in cyclic codes, which determine whether to scramble the symbols after data are modulated. Symbol scrambling becomes the external constellation points in a 16-QAM CE diagram. The simulation results showed that PAPR performance in the RM-CES-SLM technique was superior to that of the traditional SLM technique. In addition, the proposed technique has an error correction capability, and does not require the transmission of additional side information.

From Polar to Reed-Muller Codes: A Technique to Improve the Finite-Length Performance

We explore the relationship between polar and RM codes and we describe a coding scheme which improves upon the performance of the standard polar code at practical block lengths. Our starting point is the experimental observation that RM codes have a smaller error probability than polar codes under MAP decoding. This motivates us to introduce a family of codes that interpolates between RM and polar codes, call this family C-inter = {C-alpha:alpha is an element of[0;1]}, where C alpha vertical bar(alpha=1) is the original polar code, and C alpha vertical bar(alpha=0) is an RM code. Based on numerical observations, we remark that the error probability under MAP decoding is an increasing function of alpha. MAP decoding has in general exponential complexity, but empirically the performance of polar codes at finite block lengths is boosted by moving along the family C-inter even under low-complexity decoding schemes such as, for instance, belief propagation or successive cancellation list decoder. We demonstrate the performance gain via numerical simulations for transmission over the erasure channel as well as the Gaussian channel.

Spherically Punctured Biorthogonal Codes

Consider a binary Reed-Muller code RM(r,m) defined on the hypercube \BB F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> and let all code positions be restricted to the m-tuples of a given Hamming weight b. In this paper, we specify this single-layer construction obtained from the biorthogonal codes RM(1,m) and the Hadamard codes H(m). Both punctured codes inherit some recursive properties of the original RM codes; however, they cannot be formed by the recursive Plotkin construction. We first observe that any code vector in these codes has Hamming weight defined by the weight w of its information block. More specifically, this weight depends on the absolute values of the Krawtchouk polynomials K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (w). We then study the properties of the Krawtchouk polynomials and show that the minimum code weight of a single-layer code RM(1,m,b) is achieved at the minimum input weight w = 1 for any . We further refine our codes by limiting the possible weights w of the input information blocks. As a result, some of the designed code sequences meet or closely approach the Griesmer bound. Finally, we consider more general punctured codes whose positions form several spherical layers.

Reed-Muller codes and Barnes-Wall lattices: Generalized multilevel constructions and representation over GF(2 q )

Generalized multilevel constructions for binary RM(r,m) codes using projections onto GF(2 q ) are presented. These constructions exploit component codes over GF(2), GF(4),..., GF(2 q ) that are based on shorter Reed-Muller codes and set partitioning using partition chains of length-2 l codes. Using these constructions we derive multilevel constructions for the Barnes-Wall ?(r,m) family of lattices which also use component codes over GF(2), GF(4),..., GF(2 q ) and set partitioning based on partition chains of length-2 l lattices. These constructions of Reed-Muller codes and Barnes-Wall lattices are readily applicable for their efficient decoding.

Trellis based code combining protocols for Reed-Muller codes

A novel code combining system based on Reed-Muller codes is presented. Because of their simple structure RM codes are simple to decode using a trellis based soft maximum likelihood decoder (SMLD). The decoder exploits the modular structure of the RM code to construct a set of nested trellises which minimise the complexity of the decoder by re-using the results of previous decoding attempts. A protocol utilising this technique to produce an efficient code combining ARQ-scheme is also introduced.

Cyclic subcodes of generalized Reed-Muller codes

We consider certain subcodes of generalized Reed-Muller (GRM) codes, which we call homogeneous generalized Reed-Muller (HRM) codes. In general, they have a much better minimum distance than the GRM codes. The parameters of HRM codes are related to those of projective Reed-Muller (PRM) codes. Unlike most PRM codes, punctured HRM codes are cyclic. Under the trace map, HRM codes map to binary codes. These are in general much larger than classical RM codes, for the same minimum distance.

Coset partition of additive groups and minority-logic decoding algorithm for RM codes

Two new notions for the coset partition of dyadic additive groups are proposed, and their sufficient and necessary conditions are also given. On the basis of these works, the feasibility problem of implementing minority-logic decoding algorithm for RM codes is solved.

Restrictions on weight distribution of Reed-Muller codes

It is shown that in the rth order binary Reed-Muller code of length N = 2m and minimum distance d = 2m−r, the only code-words having weight between d and 2d are those with weights of the form 2d - 2i for some i. The same result also holds for certain supercodes of the RM codes.