A study of the separating property in Reed-Solomon codes by bounding the minimum distance

According to their strength, the tracing properties of a code can be categorized as frameproof, separating, IPP and TA. It is known that, if the minimum distance of the code is larger than a certain threshold then the TA property implies the rest. Silverberg et al. ask if there is some kind of tracing capability left when the minimum distance falls below the threshold. Under different assumptions, several papers have given a negative answer to the question. In this paper, further progress is made. We establish values of the minimum distance for which Reed-Solomon codes do not posses the separating property.

Program model of reed-solomon codes

Розроблена програма для моделювання завадостійких кодів Ріда-Соломона на основі об’єктно-орієнтованої технології. Вхідними даними для системи є блоки байтів для передачі через канал зв’язку, де в цих блоках можуть статися помилки. Створена програма реалізує коди типу (255,239) та (255,223) для скінченого поля з 256 елементів GF(28) зі стандартними породжуючими багаточленами x8+x4+x3+x2+1 та x8+x7+x2+x+1. Крім того, передбачена можливість у випадку необхідності додати інші типи кодів та багаточлени, які породжують скінченне поле.

NEW CONCATENATION SCHEMES BASED ON THE MULTITHRESHOLD DECODERS OF CONVOLUTIONAL SELF-ORTHOGONAL CODES FOR GAUSSIAN CHANNELS

New serial concatenation schemes based on the multithreshold decoders and di- vergent principle for the convolutional self-orthogonal codes under Gaussian channels are proposed. Using both binary and symbolic decoders on the second decoding stage of the convolutional codes are considered. Simulation results are indicated the higher performance characteristics of the proposed cascade schemes on majority decoders in comparison with clas- sical schemes based on the Viterbi algorithm and Reed-Solomon codes. A moderate increase in decoding delay during concatenation is revealed. It is determined by the absence of the need to use traditional two-dimensional concatenated structures.

On Ordinary Words of Standard Reed–Solomon Codes over Finite Fields

Reed–Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm. Usually we use the maximum likelihood decoding (MLD) algorithm in the decoding process of Reed–Solomon codes. MLD algorithm relies on determining the error distance of received word. Dür, Guruswami, Wan, Li, Hong, Wu, Yue and Zhu et al. got some results on the error distance. For the Reed–Solomon code [Formula: see text], the received word [Formula: see text] is called an ordinary word of [Formula: see text] if the error distance [Formula: see text] with [Formula: see text] being the Lagrange interpolation polynomial of [Formula: see text]. We introduce a new method of studying the ordinary words. In fact, we make use of the result obtained by Y.C. Xu and S.F. Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed–Solomon codes over the finite field of [Formula: see text] elements. This completely answers an open problem raised by Li and Wan in [On the subset sum problem over finite fields, Finite Fields Appl. 14 (2008) 911–929].

Platonic entanglement

We present a construction of highly entangled states defined on the topology of a platonic solid using tensor networks based on ancillary Absolute Maximally Entangled (AME) states. We illustrate the idea using the example of a quantum state based on AME(5,2) over a dodecahedron. We analyze the entropy of such states on many different partitions, and observe that they come on integer numbers and are almost maximal. We also observe that all platonic solids accept the construction of AME states based on Reed-Solomon codes since their number of facets, vertices and edges are always a prime number plus one.