New class of double inductive binary error correcting codes based on the class of Reed–Muller codes

An equivalence between the rank permutation codes and a new class of binary codes has been observed. A binary code may be generated by direct transformation of a permutation code. The binary codes are usually nonlinear and may be decoded by the inverse transformation and rank correlation of the equivalent permutation.

In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters. The permutation group of Reed-Muller codes is the general affine group and can be decomposed into the semi-direct product of the translation group and the general linear group. The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code. The general linear group projects the code onto codes for which we can control the useful length and the dimension. These parameters depend on the dimension of the eigenspace of the chosen element of the general linear group for the eigenvalue 1.

Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code C, identify for each dimension v, the smallest size of the support of a sub code of $C$ of dimension u, The GHW of a code are of interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use in bounding the state complexity of the trellis representation of the code. In prior work [1] by the same authors, a code-shortening algorithm was employed to derive upper bounds on the GHW of binary projective, Reed-Muller (PRM) codes. In the present paper, we derive a matching lower bound by adapting the proof techniques used originally for Reed-Muller (RM) codes by Wei in [2]. This results in a characterization of the GHW hierarchy of binary PRM codes.

Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are still under investigation. This paper covers some of the recent developments regarding the weight enumerator and the capacity-achieving properties of RM codes, as well as some of the algorithmic developments. In particular, the paper discusses the recent connections established between RM codes, thresholds of Boolean functions, polarization theory, hypercontractivity, and the techniques of approximating low weight codewords using lower degree polynomials (when codewords are viewed as evaluation vectors of degree r polynomials in m variables). It then overviews some of the algorithms for decoding RM codes. It covers both algorithms with provable performance guarantees for every block length, as well as algorithms with state-of-the-art performances in practical regimes, which do not perform as well for large block length. Finally, the paper concludes with a few open problems.

In this research work, we have developed a communication system (transmitter / receiver) to control peak to average power (PAPR) with small bit error rate (BER) for a 4G system called multicode code division multiple access (MCCDMA). Proposed communication system works on modified Reed Muller encoded data (MRMED) string. In MRMED data is first encoded with Reed Muller (RM) code. Thereafter, encoded RM message is XORed with optimal binary string, which results lower Peak to Average Power ratio (PAPR). A well-known fact is that, bit error rate (BER) is the best performance measurement tool for a communication system. To check the integrity of our communication system, we have run the simulation for monitoring BER using MRMED sequence. Simulation work conducted, with multipath Rayleigh fading, Minimum Shift Keying (MSK) modulation and several orders of RM codes. Our results show that implementing MRMED sequences of the suggested MCCDMA communication structure, returns noticeable lower BER. For instance, in case of RM(1,4), that has error improvement proficiency of 3 (three) errors , returns BER = 8.2x10−5 adopting MSK, at SNR = 12dB. Similarly, for RM(2,3), which has error improvement efficiency of 0 error and shows distinct BER of 4.9x10−4 at 12dB (SNR).In addition to using simulation for checking BER performance of our communication system, we have also shown in our results that, as the error improvement capability of different RM codes surges, correspondingly we get a lower BER.

Recently introduced Polar coding is the first practical coding technique that can be proven to achieve the Shannon capacity for a multitude of communication channels. Polar codes are close to Reed-Muller codes except the fact that they are tuned for the parameters of the channel. Hence, Polar codes are shown to offer better performance, e.g., in the erasure channel. It is known that second order Reed-Muller codes can be used for Compressed Sensing. Inspired by that result, we propose Polar codes as measurement matrices in CS and compare their numerical performances. We also present the algebraic relation between the erasure channel and CS theory, and discuss fast solution techniques.

Recently, a new class of binary codes for overloaded CDMA systems are proposed that not only has the ability of errorless communication but also suitable for detecting active users. These codes are called COWDA. In [1], a Maximum Likelihood (ML) decoder is proposed for this class of codes. Although the proposed scheme for coding/decoding shows impressive performance, the decoder can be significantly improved. In this paper, by assuming practical conditions for the traffic in the system, we suggest and prove an algorithm that increases the performance of the decoder several orders of magnitude (the Bit-Error-Rate (BER) is divided by a factor of about 400 in some E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> 's). The algorithm supposes the Poisson distribution for the time of activation/deactivation of the users.

For any odd prime power $q$ we first construct a certain non-linear binary code $C(q,2)$ having $(q^2-q)/2$ codewords of length $q$ and weight $(q-1)/2$ each, for which the Hamming distance between any two distinct codewords is in the range $[q/2-3\sqrt q/2,\ q/2+3\sqrt q/2]$ that is, 'almost constant'. Moreover, we prove that $C(q,2)$ is distance-invariant. Several variations and improvements on this theme are then pursued. Thus, we produce other classes of binary codes $C(q,n)$, $n\geq 3$, of length $q$ that have 'almost constant' weights and distances, and which, for fixed $n$ and big $q$, have asymptotically $q^n/n$ codewords. Then we prove the possibility of extending our codes by adding the complements of their codewords. Also, by using results on Artin $L-$series, it is shown that the distribution ofthe $0$'s and $1$'s in the codewords we constructed is quasi-random. Our construction uses character sums associated with the quadratic character $\chi$ of $F_{q^n}$ in which the range of summation is $F_q$. Relations with the duals of the double error correcting BCH codes and the duals of the Melas codes are also discussed.

Reed–Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a versatile decoding architecture for RM codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation (MBBP) and may use any polar or RM decoder as constituent decoders. We provide extensive error-rate performance simulations for successive cancellation (SC)-, SC-list (SCL)- and belief propagation (BP)-based constituent decoders. We furthermore compare our results to existing decoding schemes and report a near-ML performance for the RM(3,7)-code (e.g., 0.04 dB away from the ML bound at BLER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> ) at a competitive computational cost. Moreover, we provide some insights into the automorphism subgroups of RM codes and SC decoding and, thereby, prove the theoretical limitations of this method with respect to polar codes.

Reed–Muller codes are one of the most well-studied families of codes; however, there are still open problems regarding their structure. Recently, a new ring-theoretic approach has emerged that provides a rather intuitive construction of these codes. This approach is centered around the notion of basic Reed–Muller codes. We recall that Reed–Muller codes over a prime field are radical powers of a corresponding group algebra. In this paper, we prove that basic Reed–Muller codes in the case of a nonprime field of arbitrary characteristic are distinct from radical powers. This implies the same result for regular codes. Also we show how to describe the inclusion graph of basic Reed–Muller codes and radical powers via simple arithmetic equations.

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enumerators of linear codes over the ring of integers modulo 4. We also look at Gray maps from Galois rings to the finite field of prime size. We first give an application of the distance preserving Gray maps to obtain good ternary codes, and then we give an inductive algebraic construction of a distance preserving Gray map from Galois rings with the homogeneous distance to the field of prime size with the Hamming distance as well as a combinatorial construction. By using the Gray maps, we obtain some results about the weight enumerators of linear codes over the ring of dual numbers. In the last part of this work, we consider the permutation invariance of binary codes and the connection with linearity over certain rings. Among the rings considered, we have different variations and extensions of the ring of dual numbers. In this context we consider the Reed-Muller codes and answer questions about permutation invariance of Reed-Muller codes under certain permutation groups and the linearity of Reed-Muller codes over these aforementioned rings.

We classify all $q$-ary $\\Delta$-divisible linear codes which are spanned by\ncodewords of weight $\\Delta$. The basic building blocks are the simplex codes,\nand for $q=2$ additionally the first order Reed-Muller codes and the parity\ncheck codes. This generalizes a result of Pless and Sloane, where the binary\nself-orthogonal codes spanned by codewords of weight $4$ have been classified,\nwhich is the case $q=2$ and $\\Delta=4$ of our classification. As an\napplication, we give an alternative proof of a theorem of Liu on binary\n$\\Delta$-divisible codes of length $4\\Delta$ in the projective case.\n

We show that, for every ordered pair of nonnegative integers (k 1, k 2), there exists a unique (up to equivalence) one-weight Z4-linear code of type 4 k 12 k 2. We derive an upper bound and a lower bound on the highest minimum distance between some extended one-weight Z4-linear codes and the Reed-Muller codes of order 1 and same lengths.

Comparing with hard decision decoding algorithms, soft decoding has a lower probability of bit error but a higher computational complexity. As a maximum-likelihood soft decoding method, the <i>A</i>* algorithm is the most basic and widely used to minimize bit error probability. However, its average computational complexity strongly depends on a seed codeword and a heuristic function utilized during the decoding process. To efficiently reduce the computational complexity while maintaining the decoding accuracy theoretically and practically, this paper proposes an improved <i>A</i>* decoding algorithm consisting of two phases. The first phase applies the greedy list decoding to the linear block code to obtain a seed codeword. According to the seed, the second phase applies the improved <i>A</i>* algorithm to obtain the final decoding output. The heuristic function used in the <i>A</i>* algorithm is modified in two aspects: (1) use more information of partial decoded symbols to improve the accuracy of the function and (2) take advantage of Hamming distance to reduce the search space. Simulations on the <i>RM</i>(5, 2) Reed-Muller codes and [128, 64] binary extended BCH code show that this improved <i>A</i>* algorithm is more efficient in average decoding complexity than many other algorithms while maintaining the decoding accuracy.

Recently, Reed---Muller (RM) codes have been investigated for half-duplex coded-cooperative relay communication systems. RM coded-cooperative (RMCC) diversity promised considerably low encoding and decoding complexity as compared to its predecessor channel codes. In this paper, we investigate the bit error rate (BER) performance of RMCC diversity for multiple relays operating in half-duplex mode with binary phase shift keying (BPSK) and quadrature amplitude modulation such as 4-QAM. In order to improve the BER performance of 4-QAM modulation, we use well known signal space diversity (SSD) technique in the proposed system. At the destination, maximal ratio combining is used for signal detection and joint majority logic decoding is used to recover information transmitted at the destination. Monte Carlo simulations show that the RMCC diversity gains increase manifolds with inclusion of multiple relays and for higher modulation i.e., 4-QAM, SSD provides additional diversity gains over the conventional 4-QAM system. Theoretical expressions for average BER are also derived and are validated with the help of Monte Carlo simulations.