Krawtchouk polynomials: K_k(x;n,p) orthogonal w.r.t. binomial distribution B(n,p). For p=½: K_k(x)=Σ(-1)^j·C(x,j)·C(n-x,k-j). Used in coding theory: the MacWilliams transform uses Krawtchouk, connecting weight distributions of a code and its dual.
How to Use
Enter k, x, n, p:
- K_k: Polynomial value
- Table: All values
- Coding: Weight enumerator
Coding Theory
MacWilliams identity: B(x) = (1/|C|)·Σ A_j·K_j(x;n,½). This relates weight distribution {A_j} of code C to its dual. Krawtchouk polynomials are the transform kernel, analogous to Fourier for the Hamming scheme.
Properties
K_k(0;n,p) = C(n,k)(1-p)^k. Symmetry: C(n,k)p^k(1-p)^{n-k}K_j(k) = C(n,j)p^j(1-p)^{n-j}K_k(j). Three-term recurrence available.
Step-by-Step Instructions
- 1Enter parameters.
- 2Compute K_k.
- 3View table.
- 4See coding connection.
- 5Compare cases.