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A linear code is a subset of a vector space over a finite field with operations of addition and scalar multiplication. Every linear combination of codewords is also a codeword.

A parity check matrix is used in coding to determine whether a transmitted code vector has an error. It is a matrix for which the product with the code vector should be a zero vector if no error occurred.

A dual code is formed from the orthogonal complement of a linear code with respect to a particular inner product. It often serves as the basis for constructing a parity check matrix.

In coding, the rank of a matrix, such as a generator or parity check matrix, can determine the linear independence of codewords and the dimension of the code.

The Hamming distance between two codewords is the number of positions at which the corresponding symbols differ. It's used to measure the error between the sent and received code.

In coding, finite fields (also known as Galois fields) are used to construct codes such as Reed-Solomon codes that can correct burst errors.

In coding theory, the basis of a vector space also applies to the set of vectors that can generate all codewords in a code through linear combinations, which is crucial for encoding messages.

The Rank-Nullity Theorem in coding theory relates to the dimensions of a codespace, stating that the rank of a matrix plus the nullity equals the number of columns. This is critical for understanding the structure of codes.

Error correcting codes are used to detect and correct errors in data transmission. They do this by introducing redundancy through encoding data using a larger bit space than the original message.

A generator matrix is used to generate codewords for linear codes. Multiplying a message vector by the generator matrix produces the corresponding codeword.

The Singleton bound is a theoretical limit for the error correcting capability of a code, stating that the size of the codewords must be at least the length of the original message plus the maximum number of errors to be corrected.

Orthogonality in coding is used in the construction of parity check and generator matrices. It ensures that error detection is accurate by maintaining orthogonal relationships between codewords.

The minimum distance of a code is the smallest Hamming distance between all pairs of distinct codewords. It determines the error correcting capability of the code.

Syndrome decoding is used for error detection and correction. It involves computing the syndrome of a received vector and comparing it against a table of syndromes to correct errors.