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komm.BCHCode

Bose–Ray-Chaudhuri–Hocquenghem (BCH) code. For given parameters $\mu \geq 2$ and $\delta$ satisfying $2 \leq \delta \leq 2^{\mu} - 1$, a binary BCH code is a cyclic code with generator polynomial given by $$ g(X) = \mathrm{lcm} \left\{ \phi_1(X), \phi_2(X), \ldots, \phi_{\delta - 1}(X) \right\}, $$ where $\phi_i(X)$ is the minimal polynomial of $\alpha^i$, and $\alpha$ is a primitive element of $\mathrm{GF}(2^\mu)$. The parameter $\delta$ must be a Bose distance. The resulting code is denoted by $\bch(\mu, \delta)$, and has the following parameters, where $\delta = 2 \tau + 1$:

  • Length: $n = 2^{\mu} - 1$
  • Dimension: $k \geq n - \mu \tau$
  • Redundancy: $m \leq \mu \tau$
  • Minimum distance: $d \geq \delta$

Only narrow-sense and primitive BCH codes are implemented. For more details, see LC04, Ch. 6 and HP03, Sec. 5.1.

The table below lists the possible values of $\delta$ for $2 \leq \mu \leq 10$.

$\mu$ $n$ Bose distances $\delta$
$2$ $3$ $3$
$3$ $7$ $3$, $7$
$4$ $15$ $3$, $5$, $7$, $15$
$5$ $31$ $3$, $5$, $7$, $11$, $15$, $31$
$6$ $63$ $3$, $5$, $7$, $9$, $11$, $13$, $15$, $21$, $23$, $27$, $31$, $63$
$7$ $127$ $3$, $5$, $7$, $9$, $11$, $13$, $15$, $19$, $21$, $23$, $27$, $29$, $31$, $43$, $47$, $55$, $63$, $127$
$8$ $255$ $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $37$, $39$, $43$, $45$, $47$, $51$, $53$, $55$, $59$, $61$, $63$, $85$, $87$, $91$, $95$, $111$, $119$, $127$, $255$
$9$ $511$ $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $35$, $37$, $39$, $41$, $43$, $45$, $47$, $51$, $53$, $55$, $57$, $59$, $61$, $63$, $73$, $75$, $77$, $79$, $83$, $85$, $87$, $91$, $93$, $95$, $103$, $107$, $109$, $111$, $117$, $119$, $123$, $125$, $127$, $171$, $175$, $183$, $187$, $191$, $219$, $223$, $239$, $255$, $511$
$10$ $1023$ $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $33$, $35$, $37$, $39$, $41$, $43$, $45$, $47$, $49$, $51$, $53$, $55$, $57$, $59$, $61$, $63$, $69$, $71$, $73$, $75$, $77$, $79$, $83$, $85$, $87$, $89$, $91$, $93$, $95$, $99$, $101$, $103$, $105$, $107$, $109$, $111$, $115$, $117$, $119$, $121$, $123$, $125$, $127$, $147$, $149$, $151$, $155$, $157$, $159$, $165$, $167$, $171$, $173$, $175$, $179$, $181$, $183$, $187$, $189$, $191$, $205$, $207$, $213$, $215$, $219$, $221$, $223$, $231$, $235$, $237$, $239$, $245$, $247$, $251$, $253$, $255$, $341$, $343$, $347$, $351$, $363$, $367$, $375$, $379$, $383$, $439$, $447$, $479$, $495$, $511$, $1023$
Notes

Attributes:

  • mu (int) –

    The parameter $\mu$ of the BCH code.

  • delta (int) –

    The Bose distance $\delta$ of the BCH code.

Examples:

>>> code = komm.BCHCode(mu=5, delta=7)
>>> (code.length, code.dimension, code.redundancy)
(31, 16, 15)
>>> code.generator_polynomial
BinaryPolynomial(0b1000111110101111)
>>> code.minimum_distance()
7

>>> komm.BCHCode(mu=7, delta=31)
BCHCode(mu=7, delta=31)

>>> komm.BCHCode(mu=7, delta=32)
Traceback (most recent call last):
...
ValueError: 'delta' must be a Bose distance (the next one is 43)

>>> komm.BCHCode(mu=7, delta=43)
BCHCode(mu=7, delta=43)