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Encoding and Decoding of B Adjacent Codes in Cyclic Form

IP.com Disclosure Number: IPCOM000074041D
Original Publication Date: 1971-Mar-01
Included in the Prior Art Database: 2005-Feb-23
Document File: 3 page(s) / 29K

Publishing Venue

IBM

Related People

Patel, AM: AUTHOR

Abstract

The b-adjacent codes are used in special applications where errors often occur in adjacent b digits. Sequences of b binary digits can be considered as symbols in (Galois Field) GF(2/b/). Hence, any code in GF(2/b/) can be used as a b-adjacent code by interpreting the field operations properly. A method is provided of obtaining cyclic codes in b-adjacent form from the codes in GF(2/b/) specified by the roots in GF(2/bm/). In the b-adjacent form, these codes can be implemented with a shift register which is constructed from conventional binary hardware. The details are described using an example.

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Encoding and Decoding of B Adjacent Codes in Cyclic Form

The b-adjacent codes are used in special applications where errors often occur in adjacent b digits. Sequences of b binary digits can be considered as symbols in (Galois Field) GF(2/b/). Hence, any code in GF(2/b/) can be used as a b-adjacent code by interpreting the field operations properly. A method is provided of obtaining cyclic codes in b-adjacent form from the codes in GF(2/b/) specified by the roots in GF(2/bm/). In the b-adjacent form, these codes can be implemented with a shift register which is constructed from conventional binary hardware. The details are described using an example.

Let alpha be a primitive element of GF(2/b/). The code, specified by beta = alpha 2/b-1/ as a root of code polynomials, is a 2-redundant perfect code in GF(2/b/). The word length in this code is n = 2/b/ + 1 and the minimum distance is three. This cyclic code will be obtained and implemented for a b-adjacent binary application.

The minimum function of beta is the following polynomial: g(x) = 1 + ( beta + beta/2//b/) X + X/2/ (1) and beta + beta/2//b/is an element of the subfield GF(2/b/). Consider the case b = 3. The word length in this code is 2/b/ + 1 = 9. The code is specified by the root Beta = alpha/7/ where alpha is the primitive element in GF(2/b/). We obtain the nonzero elements of GF(2/6/) as power of using the primitive polynomial 1 + x + x/3/ + x/4/ + x/6/ over GF(2). These elements can also be represented by polynomials of degree 6 over GF(2), or simply by binary 6-tuples, for example: alpha(1) = 010000 alpha(18) = 111010 alpha(45) = 110001 alpha(7) = 011011 alpha(27) = 001011 alpha(54) = 011010 alpha(9) = 101011 alpha(36) = 010001 alpha(56) = 110000 alpha(63) = 100000. Note that beta = alpha(7)and that beta + beta(8) = alpha(7) + alpha(56) = alpha(9). The elements 0, alpha(9), alpha(16), alpha(27)alpha(36) alpha(45)alpha(54)alpha(63) form a subfield of 8 elements. This subfield is isomorphic to CF(2/3/), obtained by taking powers of the root B of the primitive polynomial 1 + x + x/3/. The isomorphism is given in Table 1. Table 1 Elements of CF (2/6/) Corresponding elements of CF(2/3/) 0 = 000000 0 = 000 alpha/9/ = 101011 theta = 010 alpha/18/= 001011 theta/2/ = 110 alpha/27/= 010001 theta/3/ = 011 alpha/36/= 110001 theta/4/ = 111 alpha/45/= 011010 theta/5/ = 101 alpha/54/= 100000 theta/6/ = 100 alpha/63/= 100000 theta/7/ = 100.

The generator polynomial (1) can be obtained in a usable form by substituting alpha/9/ for beta + beta/8/ and then usin...