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 (n = n 1 +n 2 , k) linear codes over GF(q) that corrects all periodic errors of order r in the first block of length n 1 and all periodic errors of order s in the second block of length n 2 and no others. Further, we extend the study to the case when the errors are in the form of periodic errors of order r (and s) or more in the two subblocks.


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  = 1, 2, 3, . . ., ( − 1) and the number of its starting positions is among the first  + 1 components.
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 [12], Dass and Tyagi explored a new type of binary (1, 2) optimal codes.Similar kind of perfect codes is also studied in [13].
Further, mathematicians also started to find codes that are opposite in nature to perfect codes.Those codes are called anti-perfect codes.In this direction, an attempt is given in paper [14] by Sharma et al.These codes correct all  errors and more and no others.
In view of these studies, this paper presents ( =  1 +  2 , ) linear optimal codes over GF() that correct all periodic errors of order  in the first block of length  1 and all periodic errors of order  in the second block of length  2 and no

Weights of Periodic Errors
In coding theory, an important criterion is to look for minimum weight and structure of weight in a group of vectors.Our following theorems (which are equivalent to Plotkin bound [17], also Theorem 4.1, Peterson and Weldon [6]) are results in that direction.The weight of a vector is considered in Hamming's sense.Lemma 2. Let   denote the total weight of all periodic errors of order  in the space of all -tuples over ().Then, where   = ⌈( − )/( + 1)⌉ ( = 0, 1, 2, . . .).
Proof.We first count the total number of periodic errors of order  with weight  in the space of all -tuples.Consider a periodic error of order .The number of positions in which periodic error of order  can occur is  0 ,  1 , . . .,   where   = ⌈( − )/( + 1)⌉ ( = 0, 1, 2, . . ., ) and  ≤   (refer to Tyagi and Das [4]).So, the total number of periodic errors of order  with weight  is given by Then, Theorem 3. The minimum weight of a periodic error of order  in the space of -tuples is at most where   = ⌈( − )/( + 1)⌉,  = 0, 1, 2, . . ..
Proof.The number of periodic errors of order  in the space of -tuples over GF() is given by By using Lemma 2, the total weight of all periodic errors of order  is given by Since the minimum weight element can have at most the average weight, an upper bound on the minimum weight of periodic errors of order  is given by During the process of transmission, periodic disturbances cause occurrence of periodic errors.But it is quite possible that all the periodic components in such periodic errors may not be affected; that is, some digits are received correctly while others get corrupted.In view of this, we have the following results for periodic errors with weight  or less (without proof).Lemma 4. Let  , denote the total weight of all periodic errors of order  which are of weight  or less ( ≤   ) in the space of all -tuples.Then, where   = ⌈( − )/( + 1)⌉,  = 0, 1, 2, . . ..

Optimal Codes
Das [18] has studied the ( =  1 + 2 , ) linear code over GF() that corrects all periodic errors of order  in the first block of length  1 and all periodic errors of order  in the second block of length  2 as follows.
The code obtained from the above matrix  as a parity check matrix is a (6 + 4, 6) linear code.This code can correct all periodic errors of order 2 in the first block of length 6 and all periodic errors of order 1 in the second block of length 4 and no others.We list in Table 1 all the error vectors and their corresponding syndromes which can be seen to be all distinct and exhaustive.

Anti-Optimal Codes
In this section, we will obtain bound on ( =  1 +  2 , ) linear code over GF() that corrects all periodic errors of order  or more in the first block of length  1 ( ≥ ( 1 − 3)/2) and all periodic errors of order  or more in the second block of length  2 ( ≥ ( 2 − 3)/2) and no other errors.Taking the bound tight, we obtain anti-optimal codes.The codes are anti-optimal codes in the sense that they correct all periodic errors of order  or more in the first block of length  1 and all periodic errors of order  or more in the second block of length  2 and no others.First we prove the following lemma.
Proof.For  ≤ 2 + 1, there will be no common errors among the periodic errors of order  or more except the single errors.
Let () be the number of periodic errors of order .Then, (refer to Tyagi and Das [4]).Therefore  (, ) =  () +  ( + 1) + ⋅ ⋅ ⋅ +  ( − 2) where Let  = 2 + 2. Since any periodic error order 2 + 1 is a periodic error of order , therefore we have Let  = 2 + 3. Since ( − 1) represents the single errors and all single errors present in periodic errors of any order, so by counting the number of periodic errors of order  or more, we take the value of  up to  − 2. Also, any periodic error of order 2 + 1 is a periodic error of order .Hence  (, ) =  () +  ( + 1) + ⋅ ⋅ ⋅ +  (2) −  ( − 1) Theorem 9.The number of parity check digits for an ( =  1 +  2 , ) linear code over () that corrects all periodic errors of order  or more in the first block of length  1 ( ≥ ( 1 − 3)/2) and all periodic errors of order  or more in the second block of length  2 ( ≥ ( 2 − 3)/2) and no other error patterns is at least where ( 1 , ) and ( 2 , ) are given in Lemma 8.
Proof.This proof is based on counting the number of errors above specific type and comparing with the available cosets in the ( =  1 +  2 , ) linear code over GF().Therefore, the total number of errors including the zero vector is Thus Hence the proof of Theorem 9 is complete.
It can be verified from the error pattern syndromes shown in Table 2.

Table 2 :
Error pattern syndromes.The number of periodic errors of order  or more in the first block of length  1 is ( 1 , ).(b)The number of periodic errors of order  or more in the second block of length  2 is ( 2 , ). table.Consider