The Properties Of The Cyclotomic Cosets And The Dimensions Of Some BCH Codes

The cyclotomic cosets are the basis to study cyclic codes.Since the properties of the cyclotomic cosets are closely related tocyclic codes, especially to narrow-sense BCH codes, many researchersfrom di?erent countries have started a extensive and profound dis-cussions about the cyclotomic cosets, representative of the cyclotomiccosets and dimensions of narrow-sense BCH codes. Although someproperties of cyclotomic cosets and representatives of cyclotomic cosetsare studied in some reference literatures, the relation between the num-ber of elements in a cyclotomic coset and the form of the representativeof the cyclotomic coset has not been mentioned, neither has the pre-cise value or a range of dimensions of some narrow-sense BCH codeswhose designed distance is under special conditions.On the basis of former researchers'studies, this paper gives therelation between the number of elements in a cyclotomic coset and theform of the representative of cyclotomic coset, and deduces the precisevalue or a range of narrow-sense BCH codes whose designed distanceis under special conditions.The paper firstly discusses the factoring of xn?1 in a finite field bythe properties of cyclotomic polynomials and gives the formula of thenumber of xn?1 factoring into irreducible polynomials, which deducesthe number of cyclotomic cosets and the representative of cyclotomiccosets.Secondly, the relation between the number of elements in a cyclo-tomic coset and the form of NDS decomposition of a representative of a cyclotomic coset has been given by the definition of nondecreasingsequence and the method to measure the representative of the cyclo-tomic coset.Finally, a range or the precise value of dimensions of some narrow-sense BCH codes, whose designed distance is under special conditions,is deduced.