+ + + +
+
+sdr.m_sequence(degree: int, poly: PolyLike | None = None, index: int = 1, output: 'decimal' = 'decimal') → ndarray[Any, dtype[int64]]
+
+sdr.m_sequence(degree: int, poly: PolyLike | None = None, index: int = 1, output: 'field' = 'decimal') → FieldArray
+

Generates a maximal-length sequence (m-sequence) from a Fibonacci linear feedback shift register (LFSR).

+
+
Parameters:
+
+
degree: int

The degree \(n\) of the LFSR.

+
+
poly: PolyLike | None = None

The feedback polynomial of the LFSR over \(\mathrm{GF}(q)\). Note, the feedback polynomial +is the reciprocal of the characteristic polynomial that defines the linear recurrence relation. +The default is None which uses the reciprocal primitive polynomial of degree \(n\) over \(\mathrm{GF}(2)\), +galois.primitive_poly(2, degree).reverse().

+
+
index: int = 1

The index \(i\) in \([1, q^{n})\) of the m-sequence. The index represents the initial state of the LFSR. +The index dictates the phase of the m-sequence. The integer index is interpreted as a polynomial over +\(\mathrm{GF}(q)\), whose coefficients are the shift register values. The default is 1, which corresponds +to the \([0, \dots, 0, 1]\) state.

+
+
output: 'decimal' = 'decimal'
output: 'field' = 'decimal'

The output format of the m-sequence.

+
    +

  • "decimal" (default): The m-sequence with decimal values in \([0, q^n)\).

    • +

  • "field": The m-sequence as a Galois field array over \(\mathrm{GF}(q^n)\).

    • +
+

+
+
+
+
Returns:
+

The length-\(q^n - 1\) m-sequence.

+
+
+

References

+ +

Examples

+

Generate a maximal-length sequence of degree-4 over \(\mathrm{GF}(2)\). Compare the sequence with index 1 to +the sequence with index 2. They are just phase shifts of each other.

+
In [1]: sdr.m_sequence(4)
+Out[1]: array([1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1], dtype=uint8)
+
+In [2]: sdr.m_sequence(4, index=2)
+Out[2]: array([0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0], dtype=uint8)
+
+
+

Generate a maximal-length sequence of degree-4 over \(\mathrm{GF}(3^2)\).

+
# Characteristic polynomial
+In [3]: c = galois.primitive_poly(3**2, 4); c
+Out[3]: Poly(x^4 + x + 5, GF(3^2))
+
+# Feedback polynomial
+In [4]: f = c.reverse(); f
+Out[4]: Poly(5x^4 + x^3 + 1, GF(3^2))
+
+In [5]: x = sdr.m_sequence(4, poly=f); x
+Out[5]: array([1, 0, 0, ..., 5, 4, 6], dtype=uint8)
+
+In [6]: x.size
+Out[6]: 6560
+
+
+
+ + + + + + + + + + +