MinimumRedundancy

MinimumRedundancy is a principle in information theory and data encoding that concerns reducing the extra bits, or redundancy, used to represent information beyond its inherent uncertainty. In this context, redundancy is the difference between the average length of encoded symbols and the source entropy, measured in bits per symbol. The goal is to design code schemes in which the expected code length is as small as possible for a given source distribution.

Most common realization is by prefix codes, where no codeword is a prefix of another, enabling instantaneous decoding. The canonical benchmark is the minimum average length among all prefix codes, achieved by Huffman coding for a known discrete source distribution. In such codes, more probable symbols receive shorter codewords, and the redundancy R = L - H is minimized. The entropy H is the theoretical lower bound on average code length.

Practical approaches include arithmetic coding, which encodes entire sequences into a fractional-bit interval and can approach the entropy limit, yielding near-zero redundancy for long messages. Other methods, such as Shannon-Fano and various adaptive schemes, trade off complexity, latency, and redundancy.

MinimumRedundancy also accounts for fixed overheads and model mismatch when probabilities are unknown or change over time, in which case the realized redundancy may exceed the ideal bound. In summary, MinimumRedundancy describes the pursuit of the most efficient encoding by aligning code lengths with source probabilities to minimize wasted bits.

See also: Huffman coding, arithmetic coding, Shannon entropy, prefix codes.