Determine All Binary Cyclic Codes Of Length 4 For Each Code

Determine all binary cyclic codes of length 4. For each code, provide: generator polynomial, check polynomial, generator matrix, parity check matrix, dimension, and minimal distance

Identify and analyze all binary cyclic codes of length 4. For each code discovered, specify the generator polynomial, the check polynomial, the generator matrix, the parity check matrix, the dimension of the code, and its minimal distance. This involves examining the polynomial representations and matrix structures associated with each cyclic code, applying the properties of cyclicity, and determining the parameters that characterize these codes. Understanding these parameters provides insight into the error-correcting capabilities and structural properties of the codes of length 4 over binary fields.

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Binary cyclic codes of length 4 are a fundamental class of linear codes with widespread applications in error detection and correction. These codes are characterized by the property that any cyclic shift of a codeword results in another codeword within the same code. Since the underlying field is binary (GF(2)), the structure of these codes is directly linked to polynomials over GF(2) and their divisibility properties.

To determine all cyclic codes of length 4, we start by examining the divisors of x^4 - 1 over GF(2). The polynomial x^4 - 1 factorizes as (x^2 + 1)^2 over GF(2), because x^4 + 1 = (x^2 + 1)^2 in GF(2). The divisors of x^4 - 1 form the generator polynomials corresponding to the codes. The possible generator polynomials are therefore 1, x + 1, x^2 + 1, x^2 + x + 1, and x^4 + 1, depending on their divisibility properties and degree constraints.

The trivial code (the whole space) is generated by 1, with parameters indicating maximum dimension. The zero code corresponds to the generator polynomial x^4 + 1, with minimal distance 4. Codes generated by degree 1 and degree 2 polynomials hold specific error-correcting parameters, which can be computed via their generator matrices (constructed from the generator polynomials) and parity check matrices (derived from check polynomials). Computing these matrices involves forming the circulant matrices associated with the polynomials and analyzing their ranks and orthogonality properties.

The minimal distance of each code hinges on the weight of the smallest non-zero codeword, which is related to the roots of the generator polynomial and the structure of the code's dual. For instance, codes generated by x + 1 can be shown to have certain error-detection capabilities, while those generated by higher-degree polynomials have different parameters. The comprehensive analysis of all such codes involves enumerating each possible divisor polynomial, constructing the matrices, and calculating parameters accordingly.

References

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