The paper discusses weight distribution of periodic errors and then the optimal case on bounds of parity check digits for (

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 [

A periodic error of order

For

For

Perfect codes are the best codes among the linear codes since the parameters satisfy the Sphere-Packing (or Hamming) bound [

By perfect codes we mean the linear codes that are capable of correcting all

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

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 [

In view of these studies, this paper presents

This paper also presents the weight structure of periodic errors in the space of

The paper is organized as follows. Section

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 [

Let

We first count the total number of periodic errors of order

Consider a periodic error of order

The minimum weight of a periodic error of order

The number of periodic errors of order

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

Let

The minimum weight of a periodic error of order

Das [

The number of parity check digits for an

Considering the equality of inequality (

We now give an example of a linear code over

By putting

Error pattern syndromes.

Error patterns | Syndromes | Error patterns | Syndromes |
---|---|---|---|

100100 0000 | 1100 | 000001 0000 | 0101 |

010010 0000 | 1001 | 000000 1010 | 0011 |

001001 0000 | 1111 | 000000 0101 | 1011 |

100000 0000 | 1000 | 000000 1000 | 0010 |

010000 0000 | 1110 | 000000 0100 | 0110 |

001000 0000 | 1010 | 000000 0010 | 0001 |

000100 0000 | 0100 | 000000 0001 | 1101 |

000010 0000 | 0111 |

In this section, we will obtain bound on

If

For

Therefore

Let

The number of parity check digits for an

This proof is based on counting the number of errors above specific type and comparing with the available cosets in the

By Lemma

The number of periodic errors of order

The number of periodic errors of order

Therefore, the total number of errors including the zero vector is

Now the equality of inequality (

For

Error pattern syndromes.

Error patterns | Syndromes | Error patterns | Syndromes |
---|---|---|---|

100 000000 | 1110 | 000 001001 | 1111 |

010 000000 | 1011 | 000 100010 | 1001 |

001 000000 | 1110 | 000 010001 | 0111 |

000 100000 | 1000 | 000 100001 | 1101 |

000 100100 | 1100 | 000 000100 | 0100 |

000 010000 | 0010 | 000 000010 | 0001 |

000 010010 | 0011 | 000 000001 | 0101 |

000 001000 | 1010 |

For

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank referees for their careful reading of the paper and for their valuable suggestions.