Even code¶
A binary linear code in which every codeword has even Hamming weight; it is doubly even when all weights are divisible by four and strictly even when even but not doubly even.
Core Idea¶
An even code is a binary linear code all of whose codewords have even Hamming weight. Because the code is linear, the condition can be checked on a generating set: sums of even-weight generators remain even. Equivalently, every codeword is orthogonal over F₂ to the all-ones vector, so the code lies inside the even-weight subspace. For a binary cyclic code, evenness corresponds to the generator polynomial having x+1 as a factor under the usual polynomial representation.
Scope of Application¶
Use even code with binary field, block length, linear code or generator matrix, weight definition, proof of parity, cyclic-polynomial convention if relevant, and strictly/doubly even subtype stated. Use even code with binary field, block length, linear code or generator matrix, weight definition, proof of parity, cyclic-polynomial convention if relevant, and strictly/doubly even subtype stated.
- Coding theory. Classifies binary linear codes.
- Error detection. Uses parity constraints.
- Self-dual codes. Studies weight divisibility.
- Lattice constructions. Relates doubly even codes to lattices.
- Combinatorics. Studies weight enumerators.
Clarity¶
Evenness is a global condition over all codewords but linearity reduces checking to generators. That shortcut fails for arbitrary nonlinear code sets. The closest near miss sets the boundary: A single-parity-check code is closest and is itself the full even-weight code, while an arbitrary even code can be any linear subcode of it.
Manages Complexity¶
Weight divisibility by four adds structure not captured by parity and interacts strongly with self-duality, weight enumerators, and lattice constructions. The central all-word property–generator shortcut tradeoff is this: Linearity makes a global condition checkable from a basis. A second even parity–four-divisibility tension matters because Both sound similar while supporting different theorems.
Abstract Reasoning¶
Use three linked moves: verify binary linear-code structure; compute weights of a generating basis; confirm every generator has even parity. As a collapse test, the case exits when one codeword has odd Hamming weight or the object is not a binary linear code under the stated identity. A fourth check is to for cyclic codes, reconcile with the x+1 factor.
Knowledge Transfer¶
Divisibility constraints on word weights transfer to lattices and designs, but binary linear-code structure delimits even codes. The nearest stopping boundary is explicit: A single-parity-check code is closest and is itself the full even-weight code, while an arbitrary even code can be any linear subcode of it. The inclusion test remains: A binary linear code is even when every vector in the code has an even number of ones. The structure no longer applies when the case exits when one codeword has odd Hamming weight or the object is not a binary linear code under the stated identity. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. It is the exact mathematical genus.
Relationships to Other Abstractions¶
Current abstraction Even code Domain-specific
Parents (1) — more general patterns this builds on
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Even code is a kind of, conditional Error-Correcting Code Domain-specific
Even-weight constraint alone guarantees parity structure and some detection; correction capability depends on the complete code's minimum distance.
Condition / exception Even-weight constraint alone guarantees parity structure and some detection; correction capability depends on the complete code's minimum distance.
Hierarchy path (1) — routes to 1 parentless root
- Even code → Error-Correcting Code → Encoding And Decoding → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Even code sits in a moderately populated region (44th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Low-Density Parity-Check Code — 0.89
- Repetition Code — 0.88
- Gray Code — 0.87
- Zyablov Bound — 0.87
- Parity-Check Matrix — 0.86
Computed from structural-signature embeddings · 2026-10-08