Linear Codes over Finite Rings
- Introduction
- Constructions
- General Linear Codes
- Simple Linear Codes
ZeroCode(R, n): Rng, RngIntElt → CodeRepetitionCode(R, n): Rng, RngIntElt → CodeZeroSumCode(R, n): Rng, RngIntElt → CodeUniverseCode(R, n): Rng, RngIntElt → CodeRandomLinearCode(R, n, k): Rng, RngIntElt, RngIntElt → Code- Example: Repetition and Zero Sum Codes
- General Cyclic Codes
- Invariants
- Subcodes
- Boolean Predicates
- New Codes from Old
- Sum, Intersection, and Dual
- Standard Constructions
DirectSum(C, D): Code, Code → CodeDirectProduct(C, D): Code, Code → CodeC1 cat C2: Code, Code → CodeExtendCode(C): Code → CodeExtendCode(C, n): Code, RngIntElt → CodePadCode(C, n): Code, RngIntElt → CodePlotkinSum(C, D): Code, Code → CodePunctureCode(C, i): Code, RngIntElt → CodePunctureCode(C, S): Code, { RngIntElt } → CodeShortenCode(C, i): Code, RngIntElt → CodeShortenCode(C, S): Code, { RngIntElt } → Code- Example: Lengths of Combined Codes
- Example: Puncturing a Code over Z4
- Codeword Operations
- Construction
- Operations
u + v: ModTupRngElt, ModTupRngElt → ModTupRngElt- u: ModTupRngElt → ModTupRngEltu - v: ModTupRngElt, ModTupRngElt → ModTupRngElta * u: RngElt, ModTupRngElt → ModTupRngEltWeight(v): ModTupRngElt → RngIntEltDistance(u, v): ModTupRngElt, ModTupRngElt → RngIntEltSupport(w): ModTupRngElt → { RngIntElt }(u, v): ModTupRngElt, ModTupRngElt → RngEltInnerProduct(u, v): ModTupRngElt, ModTupRngElt → RngEltCoordinates(C, u): Code, ModTupRngElt → [ RngFinElt ]Normalize(u): ModTupRngElt → ModTupRngEltRotate(u, k): ModTupRngElt, RngIntElt → ModTupRngEltRotate(~u, k): ModTupRngElt, RngIntEltParent(w): ModTupRngElt → ModTupRng- Example: Operations on Code Words
- Accessing Components of a Codeword
- Weight Distributions
- Weight Enumerators
- Bibliography