Does there exist a binary (10,19,3) code?

Question

The parameters (10,19,3) satisfy the singleton bound because $19 \leq 2^{10-3-1}$, the Gilbert-Varshamov Bound because $19 \geq \frac{2^{10}}{B_2(c)},$ ($B_2(c) = 56$) and the Hamming bound because $19 \leq \frac{2^{10}}{B_1(c)}$ (I calculated $t=1$ because $d \geq 2t +1$, and calculated $B_1(c) = 11$.

Even though all these bounds are satisfied, as I understand this does not mean that a code of this type exists (they are necessary but not sufficient conditions). I'm working a past exam and found this problem. I guess now I should attempt to construct a code of this type to prove it's existence? Or is there another way I'm missing, or some different test I'm overlooking? Or maybe my arithmetic is wrong.

Answer 1

This looks quite easy: take a $(15,11)$ Hamming code, and shorten it by 5 bits, so you get a $(10,6)$ binary code with distance $3$ and $M=2^6=64$ codewords (way more than the $19$ codewords you needed!).

Octave/Matlab code:

>> pkg load communications
>>
>>  [H,G]=hammgen(4);
>> G
G =

   1   1   0   0   1   0   0   0   0   0   0   0   0   0   0
   0   1   1   0   0   1   0   0   0   0   0   0   0   0   0
   0   0   1   1   0   0   1   0   0   0   0   0   0   0   0
   1   1   0   1   0   0   0   1   0   0   0   0   0   0   0
   1   0   1   0   0   0   0   0   1   0   0   0   0   0   0
   0   1   0   1   0   0   0   0   0   1   0   0   0   0   0
   1   1   1   0   0   0   0   0   0   0   1   0   0   0   0
   0   1   1   1   0   0   0   0   0   0   0   1   0   0   0
   1   1   1   1   0   0   0   0   0   0   0   0   1   0   0
   1   0   1   1   0   0   0   0   0   0   0   0   0   1   0
   1   0   0   1   0   0   0   0   0   0   0   0   0   0   1

>> G(:,5:9)=[];
>> G(1:5,:)=[]
G =

   0   1   0   1   1   0   0   0   0   0
   1   1   1   0   0   1   0   0   0   0
   0   1   1   1   0   0   1   0   0   0
   1   1   1   1   0   0   0   1   0   0
   1   0   1   1   0   0   0   0   1   0
   1   0   0   1   0   0   0   0   0   1

>> size(G)
ans =

    6   10

>> gfweight(G)
ans =  3
>>

Answer 2

This code exists, because satisfying the Gilbert-Varshamov Bound is a suficient condition for the existence of a code.