Question

Construct a BCH (7,4) code with the generator matrix G(p)=p3 +p+1. (Draw the structure of the encoder )

Answer 1

BCH codes can be defined by two parameters that are code size n and the number of errors to be corrected t

Block length: n = 2m - 1

Number of information bits: k ≥ n-m*t

Minimum distance: dmin≥2t + 1.

Generator matrix G(p)=p3 +p+1.

Generator Matrix (G) for (7, 4) Code

G is a (4 × 7) matrix

The steps involved in using G(x) to create a generator matrix (G) for a systematic code are shown below.

STEP ONE  - Creating a non-systematic generating matrix G

A suitable (4 × 7) generator matrix can be constructed by writing the generator polynomial 1011 shifted right one bit for each successive row. There are 4 rows corresponding to the choice of 4 input data bits. The 4 rows are labeled (0 to 3) for reference. While this is a valid generator polynomial, it does not generate a systematic code.

0:	1	0	1	1	0	0	0
1:	0	1	0	1	1	0	0
2:	0	0	1	0	1	1	0
3:	0	0	0	1	0	1	1

STEP TWO  - Creating a systematic generating matrix   G = [I_k|P]

A systematic generator matrix G is distinguished by having a (k × k) identity matrix, often on the left-hand side as G = [I_k|P] such that each code word includes (k) data bits followed by (n-k) parity bits. Alternatively, the format G = [P|I_k] places the identity matrix on the right hand side.

To adjust the above generator matrix for a (7,4) systematic code, the leftmost columns will be manipulated to form a (4 × 4) identity matrix. To this end, begin with the top row and add to it selected rows until the first 4-bits describe the top row of an identity matrix. The new Row 0 is formed by the sum of rows (3,2,0) refering to the labels shown above. Go down the matrix for each row until the complete generator matrix is formed.

1	0	0	0	1	0	1
0	1	0	0	1	1	1
0	0	1	0	1	1	0
0	0	0	1	0	1	1

(7,4 ) BCH ENCODER

The (7, 4) BCH Encoder is implemented with a Linear Feedback Shift Register (LFSR).

(7, 4) BCH codeword are encoded.

Complete Set of (7,4) Codewords (cyclic)

cyclic code such as this, the circular shift of a valid codeword produces another valid codeword.

For example, rotating the 7-bit codeword (01) left by one bit gives the codeword (02):

(01) = 0001011
(02) = 0010110

For an (7,4) code to be cyclic, G(x) must be a factor of x⁷ + 1 and no smaller x^N + 1.