Several Codes Related To High-density Data Storage And Optical Fiber Communication

This thesis involves several codes related to high-density data storage and optical fiber com-munication.We use powerful tools including projective geometry,elliptic curves,constacyclic codes,rational functions and r-simple matrices to construct optimal symbol-pair codes,optimal b-symbol codes and asymptotically optimal OOSPCs.In Chapter 2,we study symbol-pair codes which can protect against pair-errors in symbol-pair channels,whose outputs are overlapping pairs of symbols.The research of symbol-pair codes with the largest minimum pair-distance is interesting since such codes have the best possible error-correcting capability.A symbol-pair code attaining the maximal minimum pair-distance is called a maximum distance separable(MDS)symbol-pair code.In this chapter,we focus on constructing linear MDS symbol-pair codes over the finite field Fq.We show that a linear MDS symbol-pair code over Fq with pair-distance 5 exists if and only if the length n ranges from 5 to q2+q+1.As for codes with pair-distance 6,length ranging from q + 2 to q2,we construct linear MDS symbol-pair codes by using a configuration called ovoid in projective geometry.With the help of elliptic curves,we present a construction of linear MDS symbol-pair codes for any pair-distance d + 2 with length n satisfying 7 ? d + 2 ? n ? q +[2(?)]+ ?(q)-3,where ?(q)= 0 or 1.In Chapter 3,we consider b-symbol read channels,where the read operation is performed as a sequence of b>2 consecutive symbols.In this chapter,we establish a Singleton-type bound for b-symbol codes.Codes meeting the Singleton-type bound are called maximum distance separable(MDS)codes,and they are optimal in the sense they attain the maximal minimum b-distance.We introduce a construction method using projective geometry,and then construct several infinite families of linear MDS b-symbol codes over finite fields.The lengths of these codes have a large range.And in some sense,we completely determine the existence of linear MDS b-symbol codes over finite fields for certain parameters.In Chapter 4,we give four direct constructions for OOSPCs based on polynomials and rational functions over finite fields.We also use r-simple matrices to present a recursive construction for OOSPCs.These constructions yield new families of asymptotically optimal OOSPCs.In Chapter 5,we briefly introduce another work,namely adesign,and some other topics that are still under investigation.