To find the encoding table for the (7,4) cyclic code defined by g(p)=p^3+p^2+1, we need to first determine the generator matrix. The generator matrix is a 4 x 7 matrix whose rows are formed by the coefficients of the powers of p in the codewords generated by the polynomial g(p).
We can obtain the generator matrix by performing polynomial long division of p^3, p^2, p, and 1 by g(p):
p
____________
p^3 + p^2 + 1 | p^3 + 0p^2 + 0p + 0
| p^3 + p^2 + 1
| --------
| p^2 + 1
| p^2 + p + 1
| ---------
| p + 1
| p + 1
| -----
0
Therefore, the generator matrix is:
G = [1 0 0 0 1 1 1]
[0 1 0 0 1 0 1]
[0 0 1 0 0 1 1]
[0 0 0 1 1 1 0]
To obtain the encoding table, we simply multiply the message vector by the generator matrix:
| c1 | | 1 0 0 0 1 1 1 | | m1 |
| c2 | = | 0 1 0 0 1 0 1 | x | m2 |
| c3 | | 0 0 1 0 0 1 1 | | m3 |
| c4 | | 0 0 0 1 1 1 0 | | m4 |
The minimum distance d_min of the code is the minimum number of positions in which any two distinct codewords differ. We can use the fact that the minimum distance of a cyclic code is equal to the smallest integer d such that g(p) divides into p^d + 1 to find d_min.
In this case, we need to find the smallest integer d such that p^d + 1 is divisible by g(p) = p^3 + p^2 + 1. We can use the Euclidean algorithm to find the greatest common divisor of p^7 + 1 and g(p):
p^3 + p^2 + 1
------------------------
p^3 + 0p^2 + 0p + 0 | p^7 + 0p^6 + 0p^5 + 0p^4 + 0p^3 + 0p^2 + 0p + 1
| p^7 + 0p^6 + 0p^5 + p^4 + p^3
| ----------------------------
| 0p^6 + 0p^5 + p^4 + p^3
| 0p^6 + 0p^5 + 0p^4 + 0p^3 + 0p^2
| ------------------------
| p^4 + p^2
| p^4 + 0p^3 + p^2
| -------------
| p^3 + 1
| p^3 + 0p^2 + p
| ---------
| 1 + p
| 1 + 0p + 1
| -------
| p
Therefore, the minimum distance of the code is d_min = 4.