ON CODES OVER SOME MIXED AND SINGLE ALPHABETS AND QUANTUM CODES DERIVED FROM THEM

Abstract

Error-correcting codes are used for reliable and efficient transmission of digital information through noisy communication channels. Coding theory is a field of study concerned with the construction and analysis of error-correcting codes. The advancement of coding theory has seen elegant use of many deep mathematical results in the constructions of codes. The part of coding theory that uses such mathematical results is commonly known as Algebraic Coding Theory. In the beginning, algebraic codes were studied over finite fields and were defined as vector spaces over them. However, during 1990’s algebraic codes over finite rings also started to gain a great deal of attention. Algebraic codes over finite rings are defined as modules over finite rings. Amongst the family of codes that are extensively studied are linear codes, mainly because they have a rich algebraic structure which makes them interesting and easy to implement. One of the most studied class of linear codes is the class of cyclic codes. These codes are important from practical view point as they can be efficiently implemented. Algebraic codes have been mainly studied over single alphabets. However, recently codes over some special mixed alphabets that generalize the codes over f inite rings have seen tremendous growth. The introduction of these new alphabets having rich algebraic structures and producing promising codes has also motivated researchers to obtain quantum codes from linear and cyclic codes over them alongside the codes over single alphabets. This thesis is an outcome of our comprehensive study on codes over some mixed as well as single alphabets and deriving of quantum codes from linear or cyclic codes over such alphabets; including the codes from chain extensions of some single alphabet rings. We have introduced a new mixed alphabet ZpZp[u]/uk , uk = 0. A ZpZp[u]/uk-linear code of length (r,s), (where r,s are any positive integers) is defined to be an Rk-submodule of i Zr p × Rs k with respect to a suitable scalar multiplication, where Rk = Zp[u]/uk . Such a code may also be viewed as an Rk-submodule of Rr,s = Zp[x]/xr − 1 × Rk[x]/xs − 1 . A new Gray map φ : Rk → Zpk−1 p has been defined. For ZpZp[u]/uk-cyclic codes, we have studied submodules of Rr,s and have described different characterizations of their generators by considering two different cases (r,p) = 1, (s,p)= 1 and (r,p) = 1, (s,p) = 1. Examples illustrating the construction of ZpZp[u]/uk-cyclic codes are given, and using the newly defined Gray map, their p-ary images are obtained. The p-ary images of some of the obtained ZpZp[u]/uk-cyclic codes have the parameters of optimal codes. In addition, we have also established the MacWilliams identity for complete weight enumerators of ZpZp[u]/uk-linear codes. Spotty error-correcting codes, which are used for byte error correction in computer mem ory chips, constitute an interesting and challenging area of research. We have considered linear codes over the mixed alphabet Z2(Z2+uZ2), u2 = 0, and studied the m-spotty weight enumerators for them. A MacWilliams type identity for Hamming weight enumerators of m-spotty byte error-correcting Z2(Z2 + uZ2)-linear codes is established. Another identity from this MacWilliams type identity is derived by using a Gray map. For both identities examples are given. Pless type power moments for the spotty weight enumerators of Gray images of Z2(Z2 + uZ2)-linear codes are also established. A Griesmer type bound for the Gray images of byte error-correcting codes over Z2(Z2 + uZ2) has also been derived. To obtain new quantum codes from cyclic codes we have considered the chain ring R = Fq2[u]/uk , uk = 0. A construction of q-ary quantum codes from Hermitian self orthogonal cyclic codes over R is proposed. A Gray map from R to Fk q2 is defined. Different characterizations of cyclic codes over R are given. The structure of the duals of cyclic codes over R is also described. A necessary and sufficient condition for such codes to be self orthogonal is also obtained. Hermitian construction to Gray images of self-orthogonal cyclic codes over R is used to construct quantum codes. This construction has given many new quantum codes with parameters better than some known comparable best codes available in literature and some online databases. Examples of new quantum codes obtained by this construction is given as illustration. Two tables are also given containing new quantum codes. ii In our attempt to find new quantum codes from cyclic codes over mixed alphabets having good parameters, we have introduced the mixed alphabet Z2Z2[u]/u4 , u4 = 0. The gen erator polynomials of Z2Z2[u]/u4-cyclic codes of length (r,s), with different characteriza tions, are obtained. The structure of duals of such codes are also given under some restricted conditions. Necessary and sufficient conditions for this restricted class of Z2Z2[u]/u4-cyclic codes to contain their duals or to be self-orthogonal are obtained. A Gray map is defined and binary quantum codes are obtained by using the Calderbank-Shor-Steane (CSS) con struction on Gray images of self-orthogonal or dual containing Z2Z2[u]/u4-cyclic codes. Examples illustrating the use of above construction are given. A table containing some new binary quantum codes are also presented. Wehave also obtained quantum codes from 1-generator quasi-cyclic (QC) codes of index 2 over a finite field Fq. In this context we have introduced a variant of mixed alphabet codes over Fq and call such codes Fq-double cyclic codes. We then studied the QC codes of index 2 as a special case of Fq-double cyclic codes. The structure of duals of such QC codes are determined and a necessary and sufficient condition for such codes to be self-orthogonal is obtained. Construction of 1-generator QC codes with good minimum distance is also given. The CSS construction is used to obtain quantum codes from such self-orthogonal QC codes. Examples illustrating the proposed construction are given. Two tables of new quantum codes with good parameters obtained from such QC codes over Fq are also given. Constacyclic codes are a remarkable generalization of cyclic codes and are well known for giving many good codes. We have considered constacyclic codes of length N = nps (s is a non negative integer) over R = Fpm + uFpm, u2 = 0, where p is an odd prime and m is a positive integer. A Gray map φ : R → F2 pm is defined. From the algebraic structure of (1+βu)-constacyclic codes and their duals over R a condition for a code to be self-orthogonal is determined. Quantum codes from the self-orthogonal codes over R are obtained using the Gray images of these codes and the CSS construction.