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Showing 1–30 of 204 reference entries

Physics · Information · MeasurementScientific records

Build an inspectable physics record, test and revise a bounded cart model, preserve processing history, and plan findable data with appropriate sharing conditions.

MethodMathematics & logic · InformationClassical propositions and connectives

Classical logic combines truth-valued propositions using defined connectives.

DefinitionMathematics & logic · InformationSets, membership and basic operations

A set specifies membership without counting duplicate listings.

DefinitionMathematics & logic · InformationFunctions, domain and codomain

A function assigns exactly one output to each input in its domain.

DefinitionMathematics & logic · InformationMathematical induction

A base case and general successor step establish a predicate on all nonnegative integers.

MethodMathematics & logic · InformationComplex numbers

A complex number has real coordinates a and b and an imaginary unit satisfying i²=−1.

DefinitionMathematics & logic · Information · PhysicsLinear maps and matrix representation

A linear map preserves linear combinations and has a matrix once bases are chosen.

DefinitionMathematics & logic · InformationProbability space and axioms

A probability assigns nonnegative, normalised, countably additive weights to events.

DefinitionMathematics & logic · InformationConditional probability and independence

Conditioning restricts attention to an event of positive probability.

DefinitionMathematics & logic · InformationExpected value of a discrete random variable

Expected value weights possible values by their probabilities.

DefinitionInformation · Mathematics & logicBits and unsigned binary encoding

A bit has two possible states; a fixed-width unsigned binary word encodes an integer.

DefinitionInformation · Mathematics & logicShannon entropy of a discrete distribution

Shannon entropy measures uncertainty in a specified probability distribution.

DefinitionInformationAlgorithm and decision procedure

An algorithm specifies effective steps; a decision procedure must terminate on every permitted input.

MethodInformation · Mathematics & logicUndecidability of the halting problem

No total algorithm decides halting for every program and input in a general computation model.

Mathematical statementInformation · Mathematics & logicAsymptotic upper bounds and big O

Big O bounds eventual growth up to a fixed constant factor.

DefinitionInformationFinite-alphabet mutual information

Mutual information measures statistical dependence in bits.

DefinitionInformationNear-lossless source coding threshold

Entropy bounds asymptotic compression for an iid source.

Mathematical statementInformationDiscrete memoryless channel capacity

Capacity is the maximal asymptotically reliable rate.

Mathematical statementInformationHamming distance and error correction

Separated codewords permit bounded symbol-error recovery.

Mathematical statementInformationInternet protocol layers

Protocol layers divide communication responsibilities.

DefinitionInformationTCP reliable byte-stream transport

TCP orders bytes and retransmits detected losses.

MethodInformationEncryption, hashes and message authentication

Cryptographic primitives serve distinct security properties.

DefinitionInformationPublic-key digital signatures

A signing key and a verification key have different roles.

DefinitionInformationComputable reductions and undecidability

Reduction direction determines which impossibility transfers.

Mathematical statementInformationDecision classes P and NP

Polynomial solving and polynomial verification differ.

DefinitionInformationNP-hardness and NP-completeness

NP-completeness combines verification and reduction hardness.

DefinitionInformationBounded-error randomized computation

Random choices permit a stated probability of error.

DefinitionInformationFinite graphs and trees

Graphs represent pairwise connections; trees impose acyclicity.

DefinitionInformationData-structure interfaces and operation costs

An interface defines operations; an implementation determines costs.

MethodInformationLoop invariants and total correctness

An invariant establishes correctness only with its proof obligations.

Method
Mathematical statementSource checked

Hamming distance and error correction

Information

Separated codewords permit bounded symbol-error recovery.

Key knowledge

Hamming distance counts differing coordinates of equal-length words. A binary block code with minimum distance 2t+1 can correct up to t bit substitutions by nearest-codeword decoding and detect up to 2t substitutions by codeword membership.

Conditions and scope

  • t is a nonnegative integer and codewords have one fixed length.
  • Detection and correction are different decoding tasks; insertions and deletions are excluded.
Sources · 1

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Abstract Algebra, 16.1 Error-Detecting and Correcting CodesThomas W. JudsonCourse notesopenmathbooks.orgRead source

Evidence and review

Hamming distance, block-code definitions and Theorem 16.13, odd minimum-distance correction/detection bound.

2026-10-10 · Odd-distance Theorem 16.13 and Hamming metric.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–4 of 4 relationships

Scientific relationships · 4 / 4

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • Hamming distance counts differing coordinates of equal-length words. A binary block code with minimum distance 2t+1 can correct up to t bit substitutions by nearest-codeword decoding and detect up to 2t substitutions by codeword membership.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • Over F₂ take H=[A|Iₘ] and G=[Iₙ₋ₘ;A]. Encoding message x as c=Gx is injective, HG=0, and the code is exactly ker H. The first n−m coordinates hold information and the final m supply parity checks.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • For received r=c+e with Hc=0, syndrome Hr=He. If H has distinct nonzero columns, a single-bit error's syndrome identifies its position; flip that bit. A zero syndrome indicates a codeword, and this rule uniquely corrects every error of weight at most one.
  4. Related to
    Relationship sources · 1

    Linked statements

    • For a binary symmetric channel with crossover probability 0<p<1/2 and fixed target block error 0<epsilon<1, the best rate has normal approximation C−sqrt(V/n) Q-inverse(epsilon)+O(log(n)/n). Here C=1−h2(p) and V=p(1−p)[log2((1−p)/p)]². Capacity is the leading asymptotic limit; finite length, decoder cost and the remainder matter for actual reliability.

    A realizable code/decoder may operate below the best finite-block rate.

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