Coding‑Decoding – Advanced Study Notes
(For the Odisha Police Sub‑Inspector (SI) Reasoning Ability Test – MIT‑Level Treatment)
Goal: Equip you with a rigorous, first‑principles understanding of all coding‑decoding patterns you may encounter, and the ability to solve them systematically and confidently.
1. Foundations – What is “Coding‑Decoding”?
| Term | Formal Definition | Example |
|---|---|---|
| Encoding Function | A mapping (f: X \rightarrow Y) that transforms a symbol or number from domain (X) into a coded symbol in codomain (Y). | (f(n)=3n+1) (maps 2 → 7) |
| Decoding Function | The inverse mapping (f^{-1}: Y \rightarrow X), if it exists. | (f^{-1}(m)=\frac{m-1}{3}) |
| Code | The set of all encoded symbols ({f(x) : x \in X}). | ({4,7,10,13,\dots}) |
| Decodable | An encoding is decodable if every encoded symbol uniquely identifies a single original symbol. | A bijective (f) is always decodable. |
Key Principle: Every coding‑decoding problem is essentially about finding the hidden function (f) that maps the given “source” to the “target” sequence.
2. Common Families of Encoding Functions
| Family | Typical Form | Properties | Solving Strategy |
|---|---|---|---|
| Linear (Arithmetic) Coding | (f(n)=an+b) | Injective if (a\neq 0). | Solve for (a,b) using two or more pairs. |
| Geometric (Exponential) Coding | (f(n)=a,b^{,n}) | Requires logarithms or ratio tests. | Take logs or successive ratios. |
| Polynomial Coding | (f(n)=c_0 + c_1n + c_2 n^2 + \dots) | Higher‑order differences reveal degree. | Use finite differences to find degree, then solve system. |
| Modular Coding | (f(n)=a n + b \pmod m) | Cyclic, periodic. | Use congruences, Chinese Remainder Theorem if needed. |
| Matrix Coding | (f(\mathbf{x})=A\mathbf{x}) over (\mathbb{F}_p) | Linear transformations over finite fields. | Solve linear system via Gaussian elimination in (\mathbb{F}_p). |
| Transposition / Permutation Coding | Rearrangement of digits or letters | Deterministic bijection. | Deduce the permutation by comparing positions. |
| Substitution Coding | One‑to‑one mapping of symbols (e.g. Caesar shift) | May be non‑numeric. | Compare letter positions or use frequency analysis. |
Tip: In test questions you rarely encounter true matrix coding, but the same linear‑algebra logic applies to many “linear” codes.
3. Systematic Approach to a Coding‑Decoding Problem
| Step | Action | Rationale |
|---|---|---|
| 1. Identify Domain & Codomain | List the first few elements of source and target. | Sets the stage for pattern detection. |
| 2. Check for Injectivity | Are any target symbols repeated? | Repeated targets → not one‑to‑one → consider many‑to‑one or composite functions. |
| 3. Find a Simple Relation | Compute differences, ratios, or sums. | For linear codes, first differences are constant. |
| 4. Test Higher‑Order Patterns | Use second differences, ratios of ratios, etc. | Detects quadratic, cubic, or exponential forms. |
| 5. Consider Modulo Operations | Compute differences modulo a small integer (2,3,4,5…). | Useful if the pattern repeats. |
| 6. Verify with All Given Pairs | Plug your candidate function back in. | Ensures no mis‑fit. |
| 7. Decode | Apply the inverse (if needed) to find missing elements. | Gives the answer to the question. |
| 8. Check Edge Cases | Are there ambiguous solutions? | If yes, look for additional constraints (e.g. “positive integers only”). |
Edge‑Case Checklist
- Multiple Solutions: If more than one function fits the given data, the problem usually supplies an extra rule (e.g. “use the smallest integer function”).
- Non‑Integer Outputs: In numeric sequences, decimals or fractions usually indicate a non‑linear function or a composite function.
- Missing Middle Element: Sometimes the pattern can be broken, requiring a gap‑fill approach (e.g. using a recurrence relation).
4. Advanced Theory: Codes in the Language of Mathematics
4.1 Group Theory