Maya was a graduate student in applied algebra. Her professor had assigned problem 3.7 from Ling & Xing: “Show that the binary repetition code of length ( n ) is perfect for odd ( n ).”
She stared at the page. She knew the repetition code had codewords ( 00\ldots0 ) and ( 11\ldots1 ). She knew the Hamming bound. But how to prove perfection?
Instead of searching for a leaked solution manual, she remembered her professor’s advice: “The best solution manual is your own reasoning — verified with small cases.”
Maya wrote down ( n=3 ). The spheres of radius ( t = \lfloor (3-1)/2 \rfloor = 1 ) around each codeword:
Total covered: ( 4+4=8 = 2^3 ). Perfect. solution manual for coding theory san ling better
For ( n=5 ), ( t=2 ). Sphere size: ( \binom50 + \binom51 + \binom52 = 1+5+10=16 ). Two spheres cover ( 32 = 2^5 ) vectors. Perfect.
She generalized: Sphere size = ( \sum_i=0^(n-1)/2 \binomni ). For binary repetition codes, the two spheres are disjoint and cover the whole space because any vector is closer to ( 00\ldots0 ) or ( 11\ldots1 ) — tie impossible when ( n ) odd.
She checked the Hamming bound:
[
2 \cdot \sum_i=0^(n-1)/2 \binomni \le 2^n
]
Equality holds because the sum of binomial coefficients up to ( (n-1)/2 ) is exactly ( 2^n-1 ) (symmetry). Yes — perfect.
Maya felt a thrill. She didn’t need a solution manual. She had built understanding. Maya was a graduate student in applied algebra
Example 1 (Chapter 2, Hamming distance):
Let ( C = 0000, 1100, 0011, 1111 ).
Find minimum distance.
Example 2 (Chapter 3, Syndrome decoding):
Binary Hamming code of length 7, parity check matrix ( H ) (columns = 1..7 in binary). Received ( r = 1000000 ). Compute syndrome ( s = H r^T ) = first column of ( H ) = ( (1,0,0)^T ) (binary) = 1 in decimal. Error in position 1. Corrected ( c = 0000000 ).
If you have a specific problem from Ling & Xing you’d like explained step-by-step (without the full manual), just send it. I’ll walk you through the reasoning like Maya’s mentor might have.
It seems you're looking for the solution manual to the textbook Coding Theory: A First Course by San Ling and Chaoping Xing (often referred to as "San Ling better"). Total covered: ( 4+4=8 = 2^3 )
Here’s the direct and honest answer:
Before hunting for a solution manual, it is crucial to understand the structure of the source material. Published by Cambridge University Press, this book covers:
Each chapter ends with 20–40 problems ranging from mechanical matrix operations to proof-based theorems (e.g., proving the Singleton bound or the MacWilliams identities). The solution manual for coding theory san ling better addresses these exact problems, step by step.