Python library implementing different codes and related algorithms able to solve typical textbook exercises in Aarhus University's 'Information Theory and Coding' course.
Features included:
- Linear Block Codes
- Cyclic Codes
- BCH codes
- Reed Solomon codes
- Galois Fields
- Binary Symmetric Channel
See playground.py for an implementation of the following two examples.
Run with python3 playground.py.
Please refer to the code comments for a detailed function description.
Implementations live in coding_theory_algorithms/, for example
from coding_theory_algorithms import GaloisField, CyclicCode.
The package exports BSC, GaloisField, GF2, LinearBlockCode, CyclicCode,
BCHCode, and RSCode. Utility functions are available in their submodules.
Descriptive methods use snake_case; lecture notation includes c(), s(),
G(), H(), and PU().
With uv and Make installed, run these commands from the repository root:
uv sync --locked
uv run python playground.py
make checkmake check runs lint, formatting checks, and tests without changing source files.
Use make test, make lint, or make format-check to run individual checks.
make format applies formatting changes. make compatibility checks the local
Python matrix; see compatibility instructions for setup.
Python 3.10 or newer is required. Dependencies are recorded in uv.lock;
uv sync --locked installs them into the repository's .venv. Run commands
from the repository root; the project is not installed as a distribution.
- Finite fields
- Polynomial arithmetic
- Linear and cyclic codes
- BCH and Reed–Solomon codes
- Channels
- Euclidean helpers
- Input validation
Representations and limits are documented in these references. Exact distance calculations and exhaustive tables are intended for small textbook examples.
Construct the Galois field GF(2^5) generated by p(X) = 1 + X2 + X5, showing a table with the polynomial and binary representations of its elements.
p = np.array([1,0,1,0,0,1])
GF25 = GaloisField(p)
GF25.print_info()
Construct a GF(2^5) based on primitive
polynomial pi(X) = 1 + X^2 + X^5
Assuming pi(X) has root α, there is
pi(α) = 1 + α^2 + α^5 = 0,
then α^5 = 1 + α^2
Exp. rep. vector rep. poly. rep.
-------------------------------------------
0 [0 0 0 0 0] 0
1 [1 0 0 0 0] 1
α [0 1 0 0 0] α
α^2 [0 0 1 0 0] α^2
α^3 [0 0 0 1 0] α^3
α^4 [0 0 0 0 1] α^4
α^5 [1 0 1 0 0] 1 + α^2
α^6 [0 1 0 1 0] α + α^3
α^7 [0 0 1 0 1] α^2 + α^4
α^8 [1 0 1 1 0] 1 + α^2 + α^3
α^9 [0 1 0 1 1] α + α^3 + α^4
α^10 [1 0 0 0 1] 1 + α^4
α^11 [1 1 1 0 0] 1 + α + α^2
α^12 [0 1 1 1 0] α + α^2 + α^3
α^13 [0 0 1 1 1] α^2 + α^3 + α^4
α^14 [1 0 1 1 1] 1 + α^2 + α^3 + α^4
α^15 [1 1 1 1 1] 1 + α + α^2 + α^3 + α^4
α^16 [1 1 0 1 1] 1 + α + α^3 + α^4
α^17 [1 1 0 0 1] 1 + α + α^4
α^18 [1 1 0 0 0] 1 + α
α^19 [0 1 1 0 0] α + α^2
α^20 [0 0 1 1 0] α^2 + α^3
α^21 [0 0 0 1 1] α^3 + α^4
α^22 [1 0 1 0 1] 1 + α^2 + α^4
α^23 [1 1 1 1 0] 1 + α + α^2 + α^3
α^24 [0 1 1 1 1] α + α^2 + α^3 + α^4
α^25 [1 0 0 1 1] 1 + α^3 + α^4
α^26 [1 1 1 0 1] 1 + α + α^2 + α^4
α^27 [1 1 0 1 0] 1 + α + α^3
α^28 [0 1 1 0 1] α + α^2 + α^4
α^29 [1 0 0 1 0] 1 + α^3
α^30 [0 1 0 0 1] α + α^4
-> The non-zero elements of GF(2^5) are all roots of 1 + X^31.
-> The elements of GF(2^5) are all roots of X + X^32.
A binary linear cyclic code Ccyc(n, k) has code length n = 7 and generator polynomial g(X) = 1 + X2 + X3 + X4.
(a) Find the code rate, the generator and parity check matrices of the code in systematic form, and its Hamming distance. (b) If all the information symbols are ‘1’s, what is the corresponding code vector? (c) Find the syndrome corresponding to an error in the first information symbol, and show that the code is capable of correcting this error.
g = np.array([1,0,1,1,1])
cc = CyclicCode(g, 7)
cc.print_info()
-> Linear Block Code Cb( 7 , 3 )
-> Message length (k): 3
-> Codeword length (n): 7
-> Coding rate (R = k/n): 0.42857142857142855
-> Minimum Distance (dmin): 4
-> Error Detection Capability: 3
-> Error Correction Capability (t): 1
-> Weight Distribution (A): [0 0 0 7 0 0 0]
-> Generator Matrix (G):
[[1 0 1 1 1 0 0]
[1 1 1 0 0 1 0]
[0 1 1 1 0 0 1]]
-> Parity Check Matrix (H):
[[1 0 0 0 1 1 0]
[0 1 0 0 0 1 1]
[0 0 1 0 1 1 1]
[0 0 0 1 1 0 1]]
-> Message Codeword Table:
Messages -> Codewords
[0 0 0] [0 0 0 0 0 0 0] m(X) = 0 c(X) = 0
[1 0 0] [1 0 1 1 1 0 0] m(X) = 1 c(X) = 1 + X^2 + X^3 + X^4
[0 1 0] [1 1 1 0 0 1 0] m(X) = X c(X) = 1 + X + X^2 + X^5
[1 1 0] [0 1 0 1 1 1 0] m(X) = 1 + X c(X) = X + X^3 + X^4 + X^5
[0 0 1] [0 1 1 1 0 0 1] m(X) = X^2 c(X) = X + X^2 + X^3 + X^6
[1 0 1] [1 1 0 0 1 0 1] m(X) = 1 + X^2 c(X) = 1 + X + X^4 + X^6
[0 1 1] [1 0 0 1 0 1 1] m(X) = X + X^2 c(X) = 1 + X^3 + X^5 + X^6
[1 1 1] [0 0 1 0 1 1 1] m(X) = 1 + X + X^2 c(X) = X^2 + X^4 + X^5 + X^6
-> Parity Check Equations:
c0 = m0 ⊕ m1
c1 = m1 ⊕ m2
c2 = m0 ⊕ m1 ⊕ m2
c3 = m0 ⊕ m2
c4 = m0
c5 = m1
c6 = m2
-> Syndrome Vector Equations:
s0 = r0 ⊕ r4 ⊕ r5
s1 = r1 ⊕ r5 ⊕ r6
s2 = r2 ⊕ r4 ⊕ r5 ⊕ r6
s3 = r3 ⊕ r4 ⊕ r6
-> Standard Array:
0000000 | 1011100 1110010 0101110 0111001 1100101 1001011 0010111
-----------------------------------------------------------------
1000000 | 0011100 0110010 1101110 1111001 0100101 0001011 1010111
0100000 | 1111100 1010010 0001110 0011001 1000101 1101011 0110111
0010000 | 1001100 1100010 0111110 0101001 1110101 1011011 0000111
0001000 | 1010100 1111010 0100110 0110001 1101101 1000011 0011111
0000100 | 1011000 1110110 0101010 0111101 1100001 1001111 0010011
0000010 | 1011110 1110000 0101100 0111011 1100111 1001001 0010101
0000001 | 1011101 1110011 0101111 0111000 1100100 1001010 0010110
1100000 | 0111100 0010010 1001110 1011001 0000101 0101011 1110111
1010000 | 0001100 0100010 1111110 1101001 0110101 0011011 1000111
1001000 | 0010100 0111010 1100110 1110001 0101101 0000011 1011111
1000100 | 0011000 0110110 1101010 1111101 0100001 0001111 1010011
1000010 | 0011110 0110000 1101100 1111011 0100111 0001001 1010101
1000001 | 0011101 0110011 1101111 1111000 0100100 0001010 1010110
0101000 | 1110100 1011010 0000110 0010001 1001101 1100011 0111111
1101000 | 0110100 0011010 1000110 1010001 0001101 0100011 1111111
-> Decoding Table:
Correctable Error Patterns -> Syndromes
[0 0 0 0 0 0 0] [0 0 0 0]
[1 0 0 0 0 0 0] [1 0 0 0]
[0 1 0 0 0 0 0] [0 1 0 0]
[0 0 1 0 0 0 0] [0 0 1 0]
[0 0 0 1 0 0 0] [0 0 0 1]
[0 0 0 0 1 0 0] [1 0 1 1]
[0 0 0 0 0 1 0] [1 1 1 0]
[0 0 0 0 0 0 1] [0 1 1 1]