How to Add Binary Numbers in Cyclic Codes: Why 0101 + 1111 = 1010

In cyclic codes and other linear block codes used in digital communications, binary addition does not follow the normal carry-based binary arithmetic. Instead, it uses modulo-2 addition, also known as the bitwise XOR operation. This method is simple, fast, and widely used in error-detecting and error-correcting algorithms.

Understanding Modulo-2 (XOR) Addition

Modulo-2 addition works bit by bit, and it does not generate carries. This means each bit is added independently, using the rules:

• 0 ⊕ 0 = 0
• 0 ⊕ 1 = 1
• 1 ⊕ 0 = 1
• 1 ⊕ 1 = 0

In short, XOR outputs 1 only when the bits are different.

Example: 0101 + 1111

Let’s add the two binary numbers using modulo-2 addition:

  0101
⊕ 1111
-------
  1010

Step-by-Step Table

Bit Pair	Operation	Result
0 ⊕ 1	0 + 1 (mod 2)	1
1 ⊕ 1	1 + 1 (mod 2)	0
0 ⊕ 0	0 + 0 (mod 2)	0
1 ⊕ 0	1 + 0 (mod 2)	1

So the final answer is: 0101 ⊕ 1111 = 1010

Why This Operation Matters

Modulo-2 addition is the foundation of many systems in digital communication and coding theory, including:

• Cyclic codes
• Linear block codes
• CRC (Cyclic Redundancy Check)
• Error detection and correction
• Polynomial arithmetic over GF(2)

Conclusion

Binary addition in cyclic codes is simple once you understand that it relies on XOR instead of conventional addition. Because no carries are involved, operations are faster and ideal for communication systems. That’s why:

0101 + 1111 = 1010 (in cyclic codes)