MHB Weight and distance of a binary code

AI Thread Summary
The discussion centers on the relationship between the weight and distance of a binary code, specifically examining a set of eight codewords. Participants explore the challenge of calculating the minimum distance without manually checking all possible pairs, which can be cumbersome. One proposed method involves using the triangle inequality to estimate the distance based on the weights of the codewords. Clarification is sought regarding the notation used in the proposed formula, specifically the meaning of Ci and Cj, which refer to different codewords. The conversation highlights the complexity of efficiently determining distances in binary codes.
annie122
Messages
51
Reaction score
0
Is there a relationship between the distance and weight of a binary code?
I want to find the weight and distance of the code consisting of the codewords:

0000 0000
0010 1110
0101 1100
1011 1010
1110 0101
1100 1011
0111 0011
1001 0111

(spaces inserted for readability)

The weight can be checked easily by hand, but I don't want to check 21 distances.
Is there a quicker way?

One idea I had is d(C) \leq min(w(ci) + w(cj)), from the triangle inequality.
 
Mathematics news on Phys.org
Re: weight and distance of a binary code

Yuuki said:
Is there a relationship between the distance and weight of a binary code?
I want to find the weight and distance of the code consisting of the codewords:

0000 0000
0010 1110
0101 1100
1011 1010
1110 0101
1100 1011
0111 0011
1001 0111

(spaces inserted for readability)

The weight can be checked easily by hand, but I don't want to check 21 distances.
Is there a quicker way?

One idea I had is d(C) \leq min(w(ci) + w(cj)), from the triangle inequality.

The 'brute force procedure' for find the minimum distance in a code of size M=8 requires to check 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28 distances... ... just a little question : in Your formula what are $C_{i}$ and $C_{j}$?... Kind regards $\chi$ $\sigma$
 
The 'brute force procedure' for find the minimum distance in a code of size M=8 requires to check 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28 distances...
I said 21 because I already know the distance from the code 0000 0000 from the calculation of the weights.

... just a little question : in Your formula what are Ci and Cj?
They are two different codewords.
I should have written this there, sorry.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.

Similar threads

Back
Top