50 Years of the Golomb--Welch Conjecture

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.Comment: 28 pages, 2 figure

Quasi-Perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension

A construction of two-quasi-perfect Lee codes is given over the space ?np for p prime, p ? ±5 (mod 12), and n = 2[p/4]. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee metric do not exist for dimension n ? 3 and radius r ? 2. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between coding and graph theories

On Grid Codes

If $A_{i}$ is finite alphabet for $i=1,...,n$, the Manhattan distance is defined in $\prod_{i=1}^{n}A_{i}$. A grid code is introduced as a subset of $\prod_{i=1}^{n}A_{i}$. Alternative versions of the Hamming and Gilbert-Varshamov bounds are presented for grid codes. If $A_{i}$ is a cyclic group for $i=1,...,n$, some bounds for the minimum Manhattan distance of codes that are cyclic subgroups of $\prod_{i=1}^{n}A_{i}$ are determined in terms of their minimum Hamming and Lee distances. Examples illustrating the main results are provided

On almost perfect linear Lee codes of packing radius 2

More than 50 years ago, Golomb and Welch conjectured that there is no perfect Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved that if $C$ is linear, then the Golomb-Welch conjecture is valid for $r=2$ and $n\geq 3$. In this paper, we consider the classification of linear Lee codes with the second-best possibility, that is the density of the lattice packing of $\mathbb{Z}^n$ by Lee spheres $S(n,r)$ equals $\frac{|S(n,r)|}{|S(n,r)|+1}$. We show that, for $r=2$ and $n\equiv 0,3,4 \pmod{6}$, this packing density can never be achieved.Comment: The extended abstract of an earlier version of this paper was presented in the 12th International Workshop on Coding and Cryptography (WCC) 202

Optimal Interleaving on Tori

We study t-interleaving on two-dimensional tori, which is defined by the property that any connected subgraph with t or fewer vertices in the torus is labelled by all distinct integers. It has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. We say that a torus can be perfectly t-interleaved if its t-interleaving number – the minimum number of distinct integers needed to t-interleave the torus – meets the spherepacking lower bound. We prove the necessary and sufficient conditions for tori that can be perfectly t-interleaved, and present efficient perfect t-interleaving constructions. The most important contribution of this paper is to prove that the t-interleaving numbers of tori large enough in both dimensions, which constitute by far the majority of all existing cases, is at most one more than the sphere-packing lower bound, and to present an optimal and efficient t-interleaving scheme for them. Then we prove some bounds on the t-interleaving numbers for other cases, completing a general picture for the t-interleaving problem on 2-dimensional tori

The development of an error-correcting scheme for use with a six-tone HF modem

This thesis describes the development of an error correcting system for a H.F. modem employing 6-tone Multi-Frequency Shift Keying (MFSK) as its modulation scheme. The modulation scheme was chosen to be compatible with equipment already in service and to eliminate the need to modify the existing communications infrastructure. A convolutional code together with either Viterbi decoding or Fano decoding is chosen to provide the error correction because of the potential power of such codes and because it is possible for these combinations of code and decoding method to work with any alphabet size. To detect whether correction has been successful a Cyclic Redundancy Check (CRC) is embedded within the data block before encoding.A method of using a convolutional code to provide variable rate is presented. The method uses a systematic code so that it is possible for the scheme to have a quick look to see if the first data transmission has been received error free. A search for good codes is undertaken and the effect the alphabet size has on the code spectra discussed. It is shown that a good generator sequence for a binary code is also a good generator sequence for non-binary codes.To decode the convolutional code both the Viterbi maximum likelihood decoder and the Fano sequential decoder are studied. It is argued that the Fano sequential decoder is the better choice for this application because it makes better use of system resources which will be limited in the field equipment. It is also shown that the performance of multi-level codes is better than binary codes and that an alphabet size of around 6 is optimum.The throughput of the variable rate scheme and a number of fixed rate schemes is examined. It is shown that the variable rate scheme provides the best throughput at all data rates and that the throughput improvement at the higher data rates is greatest. The effect of interleaving is also examined and results presented.To support the variable rate scheme a protocol is developed that can be used on practical H.F. channels. The potential problems with errors on both the forward and return channel are analysed and mechanisms to deal with these built-in

Challenges and Some New Directions in Channel Coding

Three areas of ongoing research in channel coding are surveyed, and recent developments are presented in each area: spatially coupled Low-Density Parity-Check (LDPC) codes, nonbinary LDPC codes, and polar coding.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/JCN.2015.00006

Multiple Packing: Lower Bounds via Error Exponents

We derive lower bounds on the maximal rates for multiple packings in high-dimensional Euclidean spaces. Multiple packing is a natural generalization of the sphere packing problem. For any $N>0$ and $L\in\mathbb{Z}_{\ge2}$, a multiple packing is a set $\mathcal{C}$ of points in $\mathbb{R}^n$ such that any point in $\mathbb{R}^n$ lies in the intersection of at most $L-1$ balls of radius $\sqrt{nN}$ around points in $\mathcal{C}$. We study this problem for both bounded point sets whose points have norm at most $\sqrt{nP}$ for some constant $P>0$ and unbounded point sets whose points are allowed to be anywhere in $\mathbb{R}^n$. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. We derive the best known lower bounds on the optimal multiple packing density. This is accomplished by establishing a curious inequality which relates the list-decoding error exponent for additive white Gaussian noise channels, a quantity of average-case nature, to the list-decoding radius, a quantity of worst-case nature. We also derive various bounds on the list-decoding error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.Comment: The paper arXiv:2107.05161 has been split into three parts with new results added and significant revision. This paper is one of the three parts. The other two are arXiv:2211.04407 and arXiv:2211.0440

Coordinated design of coding and modulation systems

The joint optimization of the coding and modulation systems employed in telemetry systems was investigated. Emphasis was placed on formulating inner and outer coding standards used by the Goddard Spaceflight Center. Convolutional codes were found that are nearly optimum for use with Viterbi decoding in the inner coding of concatenated coding systems. A convolutional code, the unit-memory code, was discovered and is ideal for inner system usage because of its byte-oriented structure. Simulations of sequential decoding on the deep-space channel were carried out to compare directly various convolutional codes that are proposed for use in deep-space systems