A Topological View of Reed–Solomon Codes

We studied a particular class of well known error-correcting codes known as Reed–Solomon codes. We constructed RS codes as algebraic-geometric codes from the normal rational curve. This approach allowed us to study some algebraic representations of RS codes through the study of the general linear group GL(n,q). We characterized the coefficients that appear in the decompostion of an irreducible representation of the special linear group in terms of Gromov–Witten invariants of the Hilbert scheme of points in the plane. In addition, we classified all the algebraic codes defined over the normal rational curve, thereby providing an algorithm to compute a set of generators of the ideal associated with any algebraic code constructed on the rational normal curve (NRC) over an extension Fqn of Fq.

THE ATTACKERS POWER BOUNDARIES FOR TRACEABILITY OF ALGEBRAIC GEOMETRIC CODES ON SPECIAL CURVES

Broadcast encryption is a data distribution protocol which can prevent malefactor parties from unauthorized accessing or copying the distributed data. It is widely used in distributed storage and network data protection schemes. To block the socalled coalition attacks on the protocol, classes of error-correcting codes with special properties are used, namely c-FP and c-TA properties. We study the problem of evaluating the lower and the upper boundaries on coalition power, within which the algebraic geometry codes possess these properties. Earlier, these boundaries were calculated for single-point algebraic-geometric codes on curves of the general form. Now, we clarified these boundaries for single-point codes on curves of a special form; in particular, for codes on curves on which there are many equivalence classes after factorization by equality of the corresponding points coordinates relation.

On the Properties of Algebraic Geometric Codes as Copy Protection Codes

Traceability schemes which are applied to the broadcast encryption can prevent unauthorized parties from accessing the distributed data. In a traceability scheme a distributor broadcasts the encrypted data and gives each authorized user unique key and identifying word from selected error-correcting code for decrypting. The following attack is possible in these schemes: groups of c malicious users are joining into coalitions and gaining illegal access to the data by combining their keys and identifying codewords to obtain pirate key and codeword. To prevent this attacks, classes of error-correcting codes with special c-FP and c-TA properties are used. In particular, c -FP codes are codes that make direct compromise of scrupulous users impossible and c -TA codes are codes that make it possible to identify one of the a‹ackers. We are considering the problem of evaluating the lower and the upper boundaries on c, within which the L-construction algebraic geometric codes have the corresponding properties. In the case of codes on an arbitrary curve the lower bound for the c-TA property was obtained earlier; in this paper, the lower bound for the c-FP property was constructed. In the case of curves with one infinite point, the upper bounds for the value of c are obtained for both c-FP and c-TA properties. During our work, we have proved an auxiliary lemma and the proof contains an explicit way to build a coalition and a pirate identifying vector. Methods and principles presented in the lemma can be important for analyzing broadcast encryption schemes robustness. Also, the c-FP and c-TA boundaries monotonicity by subcodes are proved.

NOISE IMMUNITY OF THE ALGEBRAIC GEOMETRIC CODES

Linear block noise-immune codes constructed according to algebraic curves (algebraic geometric codes) are considered, their design properties are evaluated, algorithms of construction and decoding are studied. The energy efficiency of the transmission of discrete messages by M-ary orthogonal signals in the application of algebraic geometric codes is studied; the achievable energy gain from the use of noise-immune coding is estimated. The article shows that in discrete channels without memory it is possible to obtain a significant energy gain, which increases with the transition to long algebraic geometric codes constructed from curves with a large number of points relative to the genus of the curve. It is found that the computational complexity of implementing algebraic geometric codes is comparable to other known noise-immune codes, for example, Reed-Solomon codes and others. Thus, high energy efficiency in combination with the acceptable computational complexity of implementation confirms the prospects of algebraic geometric codes use in modern telecommunication systems and networks to improve the noise immunity of data transmission channels.