BASICS OF THE THEORY OF SYNDROME NORMS FOR REED-SOLOMON CODES

Abstract

The theory of syndrome norms (TNS) is developed for Reed-Solomon codes (RS-codes), the extension of TNS, which was developed 20 years ago for the class of the class of Bose-Chaudhuri-Hocquenghem codes (BCH-codes). RS-codes are built on non-binary alphabets, therefore it contain an extremely large variety of correctable errors in contrast to BCH-codes. To correct errors, a systematic application of automorphisms of codes is proposed. Characteristic automorphisms of RS-codes are cyclic and affine substitutions forming cyclic groups Г and A whose orders coincide with the code length. Cyclic and affine substitutions commute with each other and generate a joint АГ group. These three groups act on the space of error vectors of RS-codes, breaking this space into three types of error orbits. As a rule, these orbits are complete, that is, they contain the maximum possible number of errors. The spectra of the syndromes of error orbits are also complete. The structure of the syndrome spectrums copies the structure of the orbits themselves, which in turn copy the structure of groups of code automorphisms. The concept of the norms of the error syndrome is introduced. This is vector quantity whose coordinates are determined by all kinds of pairs of components of the syndrome. It is proved that the norm of the syndrome is invariant under the action of substitution of group Г. So the norms of syndromes are invariants of each individual Г-group. The article proves a number of proposals that reflect the basic properties of the norms of syndromes. These results form the theoretical basic of the norm error correction methods by RS-codes.