galois.BCH

class galois.BCH

A primitive, narrow-sense binary $BCH (n, k)$ code.

A $BCH (n, k)$ code is a $[n, k, d]_{2}$ linear block code with codeword size $n$ , message size $k$ , minimum distance $d$ , and symbols taken from an alphabet of size 2.

To create the shortened $BCH (n - s, k - s)$ code, construct the full-sized $BCH (n, k)$ code and then pass $k - s$ bits into encode() and $n - s$ bits into decode(). Shortened codes are only applicable for systematic codes.

Examples¶

Construct the BCH code.

In [1]: galois.bch_valid_codes(15)
Out[1]: [(15, 11, 1), (15, 7, 2), (15, 5, 3), (15, 1, 7)]

In [2]: bch = galois.BCH(15, 7); bch
Out[2]: <BCH Code: [15, 7, 5] over GF(2)>

Encode a message.

In [3]: m = galois.GF2.Random(bch.k); m
Out[3]: GF([0, 1, 0, 1, 0, 1, 1], order=2)

In [4]: c = bch.encode(m); c
Out[4]: GF([0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1], order=2)

Corrupt the codeword and decode the message.

# Corrupt the first bit in the codeword
In [5]: c[0] ^= 1

In [6]: dec_m = bch.decode(c); dec_m
Out[6]: GF([0, 1, 0, 1, 0, 1, 1], order=2)

In [7]: np.array_equal(dec_m, m)
Out[7]: True

# Instruct the decoder to return the number of corrected bit errors
In [8]: dec_m, N = bch.decode(c, errors=True); dec_m, N
Out[8]: (GF([0, 1, 0, 1, 0, 1, 1], order=2), 1)

In [9]: np.array_equal(dec_m, m)
Out[9]: True

Constructors¶

BCH(n: int, k: int, primitive_poly: PolyLike | None = None, ...): Constructs a primitive, narrow-sense binary $BCH (n, k)$ code.

String representation¶

__repr__() → str: A terse representation of the BCH code.

__str__() → str: A formatted string with relevant properties of the BCH code.

Methods¶

decode(codeword: ArrayLike, errors: False = False) → GF2
decode(codeword: ArrayLike, errors) → Tuple[GF2, integer | ndarray]: Decodes the BCH codeword $c$ into the message $m$ .

detect(codeword: ArrayLike) → bool_ | ndarray: Detects if errors are present in the BCH codeword $c$ .

encode(message: ArrayLike, parity_only: bool = False) → GF2: Encodes the message $m$ into the BCH codeword $c$ .

Properties¶

property d : int: The design distance $d$ of the $[n, k, d]_{2}$ code. The minimum distance of a BCH code may be greater than the design distance, $d_{m i n} \geq d$ .

property field : Type[FieldArray]: The FieldArray subclass for the $GF (2^{m})$ field that defines the BCH code.

property G : GF2: The generator matrix $G$ with shape $(k, n)$ .

property generator_poly : Poly: The generator polynomial $g (x)$ whose roots are roots.

property H : FieldArray: The parity-check matrix $H$ with shape $(2 t, n)$ .

property is_narrow_sense : bool: Indicates if the BCH code is narrow sense, meaning the roots of the generator polynomial are consecutive powers of $α$ starting at 1, i.e. $α, α^{2}, \dots, α^{2 t}$ .

property is_primitive : bool: Indicates if the BCH code is primitive, meaning $n = 2^{m} - 1$ .

property is_systematic : bool: Indicates if the code is configured to return codewords in systematic form.

property k : int: The message size $k$ of the $[n, k, d]_{2}$ code

property n : int: The codeword size $n$ of the $[n, k, d]_{2}$ code

property roots : FieldArray: The $2 t$ roots of the generator polynomial. These are consecutive powers of $α$ , specifically $α, α^{2}, \dots, α^{2 t}$ .

property t : int: The error-correcting capability of the code. The code can correct $t$ bit errors in a codeword.