Kasami code¶

This module implements a construction for the extended Kasami codes. The “regular” Kasami codes are obtained from truncating the extended version.

The extended Kasami code with parameters $$(s,t)$$ is defined as

$\{ v \in GF(2)^s \mid \sum_{a \in GF(s)} v_a = \sum_{a \in GF(s)} a v_a = \sum_{a \in GF(s)} a^{t+1} v_a = 0 \}$

It follows that these are subfield subcodes of the code having those three equations as parity checks. The only valid parameters $$s,t$$ are given by the below, where $$q$$ is a power of 2

• $$s = q^{2j+1}$$, $$t = q^m$$ with $$m \leq j$$ and $$\gcd(m,2j+1) = 1$$

• $$s = q^2$$, $$t=q$$

The coset graphs of the Kasami codes are distance-regular. In particular, the extended Kasami codes result in distance-regular graphs with intersection arrays

• $$[q^{2j+1}, q^{2j+1} - 1, q^{2j+1} - q, q^{2j+1} - q^{2j} + 1;$$ $$1, q, q^{2j} -1, q^{2j+1}]$$

• $$[q^2, q^2 - 1, q^2 - q, 1; 1, q, q^2 - 1, q^2]$$

The Kasami codes result in distance-regular graphs with intersection arrays

• $$[q^{2j+1} - 1, q^{2j+1} - q, q^{2j+1} - q^{2j} + 1; 1, q, q^{2j} -1]$$

• $$[q^2 - 1, q^2 - q, 1; 1, q, q^2 - 1]$$

REFERENCES:

AUTHORS:

• Ivo Maffei (2020-07-09): initial version

class sage.coding.kasami_codes.KasamiCode(s, t, extended=True)

Representation of a Kasami Code.

The extended Kasami code with parameters $$(s,t)$$ is defined as

$\{ v \in GF(2)^s \mid \sum_{a \in GF(s)} v_a = \sum_{a \in GF(s)} a v_a = \sum_{a \in GF(s)} a^{t+1} v_a = 0 \}$

The only valid parameters $$s,t$$ are given by the below, where $$q$$ is a power of 2:

• $$s = q^{2j+1}$$, $$t = q^m$$ with $$m \leq j$$ and $$\gcd(m,2j+1) = 1$$

• $$s = q^2$$, $$t=q$$

The Kasami code $$(s,t)$$ is obtained from the extended Kasami code $$(s,t)$$, via truncation of all words.

INPUT:

• s,t – (integer) the parameters of the Kasami code

• extended – (default: True) if set to True, creates an extended Kasami code.

EXAMPLES:

sage: codes.KasamiCode(16,4)
[16, 9] Extended (16, 4)-Kasami code
sage: _.minimum_distance()
4

sage: codes.KasamiCode(8, 2, extended=False)
[7, 1] (8, 2)-Kasami code

sage: codes.KasamiCode(8,4)
Traceback (most recent call last):
...
ValueError: The parameters(=8,4) are invalid. Check the documentation


The extended Kasami code is the extension of the Kasami code:

sage: C = codes.KasamiCode(16, 4, extended=False)
sage: Cext = C.extended_code()
sage: D = codes.KasamiCode(16, 4, extended=True)
sage: D.generator_matrix() == Cext.generator_matrix()
True


REFERENCES:

For more information on Kasami codes and their use see [BCN1989] or [Kas1966a], [Kas1966b], [Kas1971]

generator_matrix()

Return a generator matrix of self.

EXAMPLES:

sage: C = codes.KasamiCode(16, 4, extended=False)
sage: C.generator_matrix()
[1 0 0 0 0 0 0 0 0 1 0 0 1 1 1]
[0 1 0 0 0 0 0 0 0 1 1 0 1 0 0]
[0 0 1 0 0 0 0 0 0 0 1 1 0 1 0]
[0 0 0 1 0 0 0 0 0 0 0 1 1 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 0 1 1 0 1 1 1]
[0 0 0 0 0 0 1 0 0 0 1 1 0 1 1]
[0 0 0 0 0 0 0 1 0 1 1 1 0 0 1]
[0 0 0 0 0 0 0 0 1 1 0 1 0 0 0]


ALGORITHM:

We build the parity check matrix given by the three equations that the codewords must satisfy. Then we generate the parity check matrix over $$GF(2)$$ and from this the obtain the generator matrix for the extended Kasami codes.

For the Kasami codes, we truncate the last column.

parameters()

Return the parameters $$s,t$$ of self.

EXAMPLES:

sage: C = codes.KasamiCode(16, 4, extended=True)
sage: C.parameters()
(16, 4)
sage: D = codes.KasamiCode(16, 4, extended=False)
sage: D.parameters()
(16, 4)
sage: C = codes.KasamiCode(8,2)
sage: C.parameters()
(8, 2)