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Circular dependencies can cause many unwanted effects in software programs. Most problematic from a software design point of view is the tight coupling of the mutually dependent modules which reduces or makes impossible the separate re-use of a single module.
The Hamming(7,4) code may be written as a cyclic code over GF(2) with generator + +. In fact, any binary Hamming code of the form Ham(r, 2) is equivalent to a cyclic code, [3] and any Hamming code of the form Ham(r,q) with r and q-1 relatively prime is also equivalent to a cyclic code. [4]
Cyclic codes are not only simple to implement but have the benefit of being particularly well suited for the detection of burst errors: contiguous sequences of erroneous data symbols in messages. This is important because burst errors are common transmission errors in many communication channels , including magnetic and optical storage devices.
The generator polynomial of the BCH code is defined as the least common multiple g(x) = lcm(m 1 (x),…,m d − 1 (x)). It can be seen that g(x) is a polynomial with coefficients in GF(q) and divides x n − 1. Therefore, the polynomial code defined by g(x) is a cyclic code.
Cyclic codes are a kind of block code with the property that the circular shift of a codeword will always yield another codeword. This motivates the following general definition: For a string s over an alphabet Σ , let shift ( s ) denote the set of circular shifts of s , and for a set L of strings, let shift ( L ) denote the set of all ...
Include statements, such as #include in C/C++, using in C# and import in Java. Dependencies stated in the build system (e.g. dependency tags in Maven configuration). Examples of implicit dependencies includes: [3] Relying on specific behaviour that is not well-defined by the interface exposed. Network protocols. Routing of messages over a ...
The cyclic redundancy check (CRC) is a check of the remainder after division in the ring of polynomials over GF(2) (the finite field of integers modulo 2). That is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around.
Its generator polynomial as a cyclic code is given by f ( x ) = ∏ j ∈ Q ( x − ζ j ) {\displaystyle f(x)=\prod _{j\in Q}(x-\zeta ^{j})} where Q {\displaystyle Q} is the set of quadratic residues of p {\displaystyle p} in the set { 1 , 2 , … , p − 1 } {\displaystyle \{1,2,\ldots ,p-1\}} and ζ {\displaystyle \zeta } is a primitive p ...