Binary Codes, Decimal Codes, Gray Code, and Error Detection / Correction

What are Binary Codes? (In Simple Words)

Computers only understand electrical 0s and 1s.

• But as humans, we need to type our names, college roll numbers, symbols (@, #), and phone numbers.
• A Binary Code is simply a agreed-upon pattern of 0s and 1s used to represent a specific character or number.

💡 Real-Life Indian Student Analogy:
When you sit for a university exam, DAVV gives you a unique Roll Number.
The university database doesn’t search you by your nickname; it identifies you by that code.
Binary codes do the exact same thing for computer data!

1. Decimal Codes: BCD and Excess-3

A. BCD (Binary Coded Decimal / 8421 Code)

• A weighted code where each bit has a positional weight of 8, 4, 2, and 1.
• Each decimal digit (0 to 9) is represented using 4 bits:
  • 0 = 0000, 5 = 0101, 9 = 1001.
• Invalid BCD States: Numbers from 10 (1010₂) to 15 (1111₂) are never allowed in standard BCD.

B. Excess-3 Code (XS-3)

• A non-weighted code made by adding 3 (0011₂) to each BCD digit.
  • Formula: Excess-3 = BCD + 0011₂
  • Decimal 0 → 0000 + 0011 = 0011
  • Decimal 9 → 1001 + 0011 = 1100
• Self-Complementing Property:
  If you flip all bits (1’s complement) of an Excess-3 number, you directly get the Excess-3 code of its 9’s complement!
  (For example: Excess-3 for 2 is 0101. Invert all bits to get 1010, which is the exact Excess-3 code for 7!).
  This made building early subtraction hardware very simple.

2. Gray Code (Unit Distance Code)

The Gray Code is a non-weighted code where only one single bit changes when going from one number to the next.

Why Do We Need Gray Code?

In standard binary, going from 3 to 4 changes 3 bits at the exact same moment:

• 011 → 100 (all three bits flip).
• If one switch flips slightly faster than the others, the computer might read wrong intermediate data (like 010 or 111).
• In Gray code, only 1 bit ever changes at any step. This prevents errors in robotic arms, volume knobs, and motor angle sensors.

A. Binary to Gray Code Conversion

1. The first bit (MSB) stays the same: G₃ = B₃.
2. For every next bit, do an XOR (⊕) between the current binary bit and the bit on its left:
   • Gᵢ = Bᵢ₊₁ ⊕ Bᵢ
   • (Remember XOR rule: Same bits give 0, different bits give 1).

Example: Convert Binary (1011)₂ to Gray:

• G₃ = B₃ = 1
• G₂ = 1 ⊕ 0 = 1
• G₁ = 0 ⊕ 1 = 1
• G₀ = 1 ⊕ 1 = 0
• Answer = 1110

B. Gray to Binary Code Conversion

1. The first bit (MSB) stays the same: B₃ = G₃.
2. For each next bit, XOR the newly calculated binary bit with the next Gray bit:
   • Bᵢ = Bᵢ₊₁ ⊕ Gᵢ

3. Character Codes: ASCII vs. EBCDIC

Computers need standard tables to recognize alphabets and keyboard symbols.

4. Error Detecting and Correcting Codes

When data travels through cables or Wi-Fi, electrical noise can flip a 0 into a 1 or a 1 into a 0.

A. Parity Bit (Error Detection Only)

A single extra check bit is added to the data:

• Even Parity: The total number of 1s (including parity) must be Even.
• Odd Parity: The total number of 1s must be Odd.
• Limitation: It only catches an odd number of errors (1, 3, 5). If 2 bits flip together, parity fails to notice the error!

B. Hamming Code (Detection + Auto Correction)

Invented by Richard Hamming, this code does two things:

1. Detects that an error occurred.
2. Identifies the exact bit location so the receiver can immediately flip it back to fix it!

• Parity Bit Positions: Parity bits (P₁, P₂, P₄, P₈, …) sit at powers of 2 (positions 1, 2, 4, 8).
• Data Bits: Data bits (D₃, D₅, D₆, D₇, …) fill the remaining slots.
• Syndrome Word: The receiver recalculates parity bits. The resulting binary value (the Syndrome) gives the decimal number of the faulty bit. If syndrome is 000, there are no errors!

💡 DAVV Exam Preparation Tips

5-Mark Question: “What is Gray Code? Convert binary (11011)₂ into Gray Code and explain why it is called a unit-distance code.”

• Define unit-distance code: only 1 bit flips between consecutive numbers.
• Show the step-by-step XOR calculation:
  • G₄ = 1
  • G₃ = 1 ⊕ 1 = 0
  • G₂ = 1 ⊕ 0 = 1
  • G₁ = 0 ⊕ 1 = 1
  • G₀ = 1 ⊕ 1 = 0
  • Final Gray Code = 10110.
• Mention motor position sensors as a key practical use.

10-Mark Question: “Explain error detecting and correcting codes. How does a 7-bit Hamming code detect and correct a single-bit error? Illustrate with an example.”

• Explain why parity bits fail when 2 bits flip.
• State the Hamming parity formula: 2ᵖ ≥ m + p + 1.
• Draw the 7-bit layout table (P₁, P₂, D₃, P₄, D₅, D₆, D₇).
• Show an example where bit 5 flips, and demonstrate how the syndrome bits calculate to 101₂ = 5, pointing directly to the faulty bit.