Define symbols for code length, rate, entropy, capacity, and error probability.
Core notation for Coding Rate and Channel Capacity
In information theory, the notation usually represents random variables, alphabets, distributions, code lengths, channels, and logarithm bases. The table below gives a compact starting set for coding rate and channel capacity; define any local variation before the first calculation.
| Concept | Notation | How to read it |
|---|---|---|
| Chain rule | H(X,Y)=H(X)+H(Y\mid X) | joint entropy decomposition |
| Entropy | H(X)=-\sum_x p(x)\log p(x) | average uncertainty of X |
| Conditional entropy | H(X\mid Y) | uncertainty remaining after observing Y |
| Mutual information | I(X;Y)=H(X)-H(X\mid Y) | shared information between X and Y |
Practical workflow
Start from H(X,Y)=H(X)+H(Y\mid X) and write one sentence that says it means “joint entropy decomposition.” List the objects and assumptions, evaluate a small example, and then move the verified source into the target document or codebase.
Decisions that must be explicit
- State the logarithm base and resulting unit.
- Distinguish entropy, cross-entropy, and kl divergence.
- Specify discrete sums versus continuous integrals.
Failure checks
- Assuming kl divergence is symmetric.
- Mixing bits and nats in one calculation.
- Using 0 log 0 without the standard limiting convention.
Accessibility and portability
Keep the coding rate and channel capacity source selectable and editable. For an isolated character in coding rate and channel capacity, Unicode text may be sufficient; for structured expressions, preserve LaTeX, MathML, or a native equation object. When an image of coding rate and channel capacity is unavoidable, describe the operation, inputs, conditions, and conclusion rather than listing glyph names.
Verification checklist
- Verify probabilities normalize.
- Test deterministic and uniform distributions.
- Confirm the order of p and q in every divergence.
- Confirm every symbol used in coding rate and channel capacity has one defined meaning in the local context.
- Reopen the exported file for Coding Rate and Channel Capacity and compare it with the editable source.
How this guide was checked
Page purpose: coding rate capacity notation — Understand and apply the topic in mathematical or scientific writing
Automated quality check: Kept noindex until critical findings are resolved.
Verification references
These primary standards and official documentation pages were used to check character identity, syntax, or platform behavior described above.
- The Unicode StandardUnicode Consortium — Character identity, encoding, names, and conformance.
- Unicode Technical Report #25: Unicode Support for MathematicsUnicode Consortium — Mathematical character usage, variants, and notation support.
- LaTeX Project DocumentationThe LaTeX Project — LaTeX syntax, authoring model, and official documentation links.
- MathML CoreW3C — Semantic web mathematics elements and browser behavior.