Hamming codes¶

Given an integer $$r$$ and a field $$F$$, such that $$F=GF(q)$$, the $$[n, k, d]$$ code with length $$n=\frac{q^{r}-1}{q-1}$$, dimension $$k=\frac{q^{r}-1}{q-1} - r$$ and minimum distance $$d=3$$ is called the Hamming Code of order $$r$$.

REFERENCES:

class sage.coding.hamming_code.HammingCode(base_field, order)

Representation of a Hamming code.

INPUT:

• base_field – the base field over which self is defined.
• order – the order of self.

EXAMPLES:

sage: C = codes.HammingCode(GF(7), 3)
sage: C
[57, 54] Hamming Code over GF(7)
minimum_distance()

Return the minimum distance of self.

It is always 3 as self is a Hamming Code.

EXAMPLES:

sage: C = codes.HammingCode(GF(7), 3)
sage: C.minimum_distance()
3
parity_check_matrix()

Return a parity check matrix of self.

The construction of the parity check matrix in case self is not a binary code is not really well documented. Regarding the choice of projective geometry, one might check:

• the note over section 2.3 in [Rot2006], pages 47-48
• the dedicated paragraph in [HP2003], page 30

EXAMPLES:

sage: C = codes.HammingCode(GF(3), 3)
sage: C.parity_check_matrix()
[1 0 1 1 0 1 0 1 1 1 0 1 1]
[0 1 1 2 0 0 1 1 2 0 1 1 2]
[0 0 0 0 1 1 1 1 1 2 2 2 2]