Coding‑Decoding – Advanced Study Notes
Reasoning Ability – Assam TET
(MIT‑Level Treatment from First Principles)
1. What is Coding‑Decoding?
| Aspect | Definition |
|---|---|
| Coding | The process of replacing each element of a given set (letters, numbers, symbols, words, etc.) with another element according to a fixed rule. |
| Decoding | The inverse process: restoring the original element from its coded form. |
| Purpose | Test logical reasoning, pattern recognition, and the ability to work with bijective (one‑to‑one) or many‑to‑one mappings. |
Key Insight – In most exam‑style problems, the mapping is bijective (no two different inputs map to the same output).
If the mapping is not bijective, the problem is under‑determined, and extra clues are needed.
2. Mathematical Foundations
2.1 Set Theory & Functions
Let
- A be the set of original elements
- B be the set of coded elements
A coding is a function f : A → B.
- Injective: every distinct element of A maps to a distinct element of B.
- Surjective: every element of B is an image of some element of A.
- Bijective: both injective and surjective.
Why bijective?
Bijectivity guarantees a unique decoding. If f is not injective, two different inputs could produce the same coded output, leading to ambiguity.
2.2 Permutations & Compositions
- A coding that rearranges symbols (e.g., ‘A→C’, ‘B→B’, ‘C→A’) is a permutation of the alphabet.
- Composite codings: applying two codings sequentially is equivalent to a single coding that is the composition of the two functions.
2.3 Graph Representation
A coding can be visualized as a directed graph:
A ──► B
│ ▲
└─────┘
- Each edge represents the mapping rule.
- Cycles indicate repeated patterns.
- Disconnected components highlight independent sub‑mappings.
3. Types of Coding‑Decoding Problems
| Type | Typical Pattern | Examples |
|---|---|---|
| 1. Substitution | Replace each symbol with another fixed symbol. | “A→C, B→B, C→A” |
| 2. Positional Shift | Shift symbols by a fixed number of places (e.g., Caesar cipher). | “A→D, B→E, …” |
| 3. Arithmetic Operations | Apply a mathematical operation to each symbol’s numeric value. | “1→4, 2→9, 3→16” (square) |
| 4. Reversal / Mirror | Reverse the order of symbols or digits. | “1234 → 4321” |
| 5. Grouping / Chunking | Map a group of symbols to a single symbol. | “AB → X, CD → Y” |
| 6. Concatenation / Interleaving | Combine two sequences. | “ABCD → A1B2C3D4” |
| 7. Mixed / Hybrid | Combine several of the above. | “A→C, B→E, (C+D)→F” |
4. Strategy for Solving Coding‑Decoding Questions
-
Identify the Domain & Codomain
- List all possible symbols in A and B.
- Check for missing or extra symbols (edge cases).
-
Check for Bijectivity
- Are all outputs distinct?
- If not, look for hints that force a unique decoding.
-
Look for Patterns
- Linear:
f(x) = ax + b - Quadratic:
f(x) = ax² + bx + c - Modulo:
f(x) = (x + k) mod n - Permutation cycles: Identify cycles of length 2, 3, etc.
- Linear:
-
Test with Examples
- Plug given pairs into a hypothesised rule.
- Verify consistency across all pairs.
-
Edge Cases
- Repeated mapping: Two different inputs map to the same output.
- Unmapped symbols: Not all symbols appear in the given pairs.
- Non‑numeric symbols: Convert to numeric equivalents if needed.
-
Decode
- Once the rule is established, invert the function to retrieve the original message.
5. Common Edge Cases & How to Handle Them
| Edge Case | Impact | Resolution |
|---|---|---|
| Missing pairs | Insufficient data to infer the rule uniquely. | Seek additional clues in the problem statement or assume standard conventions (e.g., alphabetical order). |
| Non‑bijective mapping | Ambiguous decoding. | Look for constraints (e.g., “no symbol repeats”) or treat the problem as a partial decoding. |
| Multiple possible codings | Several functions satisfy given pairs. | Use additional conditions such as “maximum possible value” or “smallest possible alphabet size”. |
| Symbols with different domains | Mixing letters, digits, and symbols. | Map each domain to a number (e.g., A=1, B=2, …, 0=10) before applying numeric operations. |
| Large alphabets | Risk of combinatorial explosion. | Reduce the problem by focusing on relative differences (e.g., difference between successive values). |
6. Advanced Theory: Coding as a Group Action
- The set of all bijective codings on