Channel Coding Ontology

FLOW ALGEBRA
TITLE: Channel Coding Ontology
DOMAIN: Discrete Mathematics / Coding Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/coding-theory/channel-coding-ontology.html
VERSION: 1.0
DATE: 2026-10-11
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
SOURCE_STATUS: Mathematically grounded synthesis; live page not independently verified.

1. PURPOSE
Represent channel coding as typed mathematical transformations from information symbols into redundant codewords transmitted through noisy channels, with explicit decoding, reliability and validation constraints.

2. TYPED ENTITIES
Message: element of message alphabet M^k.
Alphabet: finite symbol set A, cardinality q >= 2.
Code: subset C of A^n, with blocklength n.
Encoder: map E: M^k -> C (injective for lossless message recovery).
Codeword: c = E(m), c in C.
Channel: conditional distribution W(y|x) on input and output alphabets.
ReceivedWord: y produced by channel given c.
Decoder: D: Y^n -> M^k or M^k union {failure}.
Rate: R = (log2 |M^k|)/n bits per channel use when n uses are made.
MinimumDistance: d_min = min_{c != c' in C} d_H(c,c') for |C| >= 2.
ErrorPattern: deviation between transmitted and received symbols.
Syndrome: H y^T for a linear code over a finite field, with compatible dimensions.
Capacity: C(W), supremum of achievable reliable information rates for the specified channel model.

3. CORE RELATIONSHIPS
Message --ENCODED_BY--> Encoder
Encoder --PRODUCES--> Codeword
Codeword --BELONGS_TO--> Code
Codeword --TRANSMITTED_THROUGH--> Channel
Channel --PRODUCES--> ReceivedWord
ReceivedWord --DECODED_BY--> Decoder
Decoder --ESTIMATES--> Message
Code --HAS_RATE--> Rate
Code --HAS_MINIMUM_DISTANCE--> MinimumDistance
Channel --HAS_CAPACITY--> Capacity

4. FLOW OPERATIONS
flow ChannelCoding(m, E, W, D):
  require m in domain(E)
  c := E(m)
  require c in C
  y ~ W(. | c)
  m_hat := D(y)
  return {message:m, codeword:c, received:y, decoded:m_hat,
          success:(m_hat == m)}

flow ValidateCode(C, n, A):
  require C subset_of A^n
  require |C| >= 1
  return {valid:true, blocklength:n, size:|C|}

flow DistanceDecode(y, C):
  distances := {c -> d_H(y,c) for c in C}
  candidates := argmin(distances)
  if |candidates| != 1: return failure
  return unique(candidates)

5. ALGEBRAIC LAWS AND GUARANTEES
For a q-ary block code with M = |C|: symbol rate R_q = log_q(M)/n.
For a binary (n,k) linear code with dimension k: |C| = 2^k and R = k/n bits per channel use.
For a code of minimum Hamming distance d_min:
  guaranteed detectable symbol errors <= d_min - 1.
  guaranteed uniquely correctable symbol errors t = floor((d_min - 1)/2).
For a linear code C with parity-check matrix H: c in C iff H c^T = 0.
Syndrome of y=c+e is H y^T = H e^T over the relevant finite field.
Singleton bound for q-ary code: M <= q^(n-d_min+1).
For a q-ary linear [n,k,d] code: d <= n-k+1.
Channel capacity is model-dependent; reliable transmission is asymptotically achievable below capacity under the applicable coding theorem, not guaranteed for any fixed finite code.

6. INFERENCE RULES
IF a code has d_min >= 2t+1 THEN every error pattern of Hamming weight <= t has a unique nearest transmitted codeword.
IF a received word has zero syndrome THEN it satisfies the linear code parity checks; this does NOT establish that no transmission error occurred.
IF the code rate exceeds channel capacity THEN arbitrarily reliable transmission is impossible under the standard memoryless-channel asymptotic model.
IF the decoder returns failure THEN recovery is unconfirmed, not necessarily impossible.
IF the channel is noiseless AND E is injective AND D is its left inverse on C THEN D(E(m)) = m.

7. SEMANTIC_EQUIVALENCE
channel code ~= error-control code (in error-control context; not all source codes)
codeword ~= encoded block (when referring to a single block-code output)
code rate ~= information symbols per channel symbol (only with a specified common alphabet)
error detection != error correction
channel coding != source coding
zero syndrome != proof of error-free reception

8. VALIDATION
V1: Check encoder input/output types and injectivity.
V2: Check all codewords have length n and symbols in A.
V3: Check W(y|x) >= 0 and sum_y W(y|x) = 1 for each x in discrete channels.
V4: Verify decoder output type and declared failure handling.
V5: Verify code rate units and alphabet assumptions.
V6: Verify d_min and claimed correction/detection guarantees.
V7: Verify generator/parity-check matrix dimensions and rank where relevant.
V8: Never infer practical reliability solely from an asymptotic capacity theorem.

9. CONFIDENCE
Mathematical definitions and stated standard bounds: HIGH under their explicit assumptions.
Specific code/channel performance: UNKNOWN until channel parameters and decoding method are supplied.
Live-page correspondence: UNVERIFIED.

10. FLOW_HASH
Canonicalise title, source URL, typed entities, relationships, operations, laws, assumptions and validation rules as UTF-8; compute SHA-256 on the canonical representation at ingestion. No precomputed hash is asserted here.

11. UPDATE_MODEL
On update: preserve version history; compare definitions, type signatures, laws, proof assumptions, examples and source revisions; invalidate dependent claims if assumptions change; regenerate FLOW_HASH.

12. AI_CONSUMPTION
Expose typed entities and relations for semantic retrieval.
Use explicit assumptions before applying distance or capacity claims.
Return traceable intermediate outputs for encoding, channel simulation and decoding.
Distinguish deterministic code properties from probabilistic channel performance.
Require validation and confidence checks before producing operational recommendations.

END FLOW ALGEBRA