Weight Distribution of Periodic Errors and Optimal/Anti-Optimal Linear Codes

The paper discusses weight distribution of periodic errors and then the optimal case on bounds of parity check digits for ( , ) linear codes over that corrects all periodic errors of order in the first block of length and all periodic errors of order in the second block of length and no others. Further, we extend the study to the case when the errors are in the form of periodic errors of order (and ) or more in the two subblocks. 1. Introduction In coding theory, many types of error patterns have been considered, and codes accordingly are constructed to combat such error patterns. Periodic errors are one type of error patterns that are found in channels like astrophotography [1], gyroscope and computed tomography [2]. Such error occurs due to happening of disturbances periodically. So, there is a need to study such errors and to develop codes dealing with such errors. It was in this spirit that codes detecting/correcting such errors were studied by Das and Tyagi [3, 4]. A periodic error of order is defined as follows. Definition 1. A periodic error of order is a vector whose nonzero components are located at shifting positions in a code vector where and the number of its starting positions is among the first components. For , the periodic errors of order 1 are the vectors where error may occur in 1st, 3rd, 5th, , positions or 2nd, 4th, 6th, …, positions. For example, in a vector of length 8, periodic errors of order 1 are of the type 10101000, 00101000, 0010101, 10101010, 10001010, 01010101, 01000101, 00000101, 00000001, and so forth. For , the periodic errors of order 2 may look like 10010010, 10000010, 00010010, 01001001, 01000001, 01000000, 00001001, and so forth in a vector of length 8. Perfect codes are the best codes among the linear codes since the parameters satisfy the Sphere-Packing (or Hamming) bound [5, 6]. It was a big challenge for mathematician to search for such codes for several years in the past. It was finally established that there are no perfect codes other than the single error correcting Hamming [5] codes, double and triple error correcting Golay codes [7], and the Repetitive codes (refer to Tietavainen [8], Tietavainen and Perko [9], and van Lint [10]). By perfect codes we mean the linear codes that are capable of correcting all t or fewer errors and no others. Thereafter several attempts were given to find codes that are not perfect in the usual sense but that correct certain type of error pattern and no more. Such codes are called optimal codes. Sharma and Dass [11] were the first who attempted to find such codes. In paper