Underbound length of a ternary linear code












0












$begingroup$


I have to prove the following theorem:



Let $C$ be a ternary linear $[n,2,d]$-code. Prove that $d+ d/3 leq n$.



Now I don't really see what the best approach for this would be. I was thinking maybe the generator matrix, but I didn't really get anywhere when trying this. Any tips?






coding-theory







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  • $begingroup$
    $d+ d/3 leq n$ looks weird, as it would be simpler to write $ frac43 d leq n$ Are you sure you got it right?
    $endgroup$
    – leonbloy
    Jan 8 at 20:47












  • $begingroup$
    Yes I'm sure...
    $endgroup$
    – xzeo
    Jan 9 at 11:08










  • $begingroup$
    Hint: consider the possible columns of $G$ and how many linear combinatios produce a zero in each position....
    $endgroup$
    – leonbloy
    Jan 9 at 13:43






  • 2




    $begingroup$
    This is a special case of the Griesmer bound. The 2-dimensional case can, indeed, be proven the way leonbloy indicated. But the general result isn't very difficult either.
    $endgroup$
    – Jyrki Lahtonen
    Jan 9 at 15:25
















0












$begingroup$


I have to prove the following theorem:



Let $C$ be a ternary linear $[n,2,d]$-code. Prove that $d+ d/3 leq n$.



Now I don't really see what the best approach for this would be. I was thinking maybe the generator matrix, but I didn't really get anywhere when trying this. Any tips?






coding-theory







share|cite|improve this question









$endgroup$












  • $begingroup$
    $d+ d/3 leq n$ looks weird, as it would be simpler to write $ frac43 d leq n$ Are you sure you got it right?
    $endgroup$
    – leonbloy
    Jan 8 at 20:47












  • $begingroup$
    Yes I'm sure...
    $endgroup$
    – xzeo
    Jan 9 at 11:08










  • $begingroup$
    Hint: consider the possible columns of $G$ and how many linear combinatios produce a zero in each position....
    $endgroup$
    – leonbloy
    Jan 9 at 13:43






  • 2




    $begingroup$
    This is a special case of the Griesmer bound. The 2-dimensional case can, indeed, be proven the way leonbloy indicated. But the general result isn't very difficult either.
    $endgroup$
    – Jyrki Lahtonen
    Jan 9 at 15:25














0












0








0





$begingroup$


I have to prove the following theorem:



Let $C$ be a ternary linear $[n,2,d]$-code. Prove that $d+ d/3 leq n$.



Now I don't really see what the best approach for this would be. I was thinking maybe the generator matrix, but I didn't really get anywhere when trying this. Any tips?






coding-theory







share|cite|improve this question









$endgroup$




I have to prove the following theorem:



Let $C$ be a ternary linear $[n,2,d]$-code. Prove that $d+ d/3 leq n$.



Now I don't really see what the best approach for this would be. I was thinking maybe the generator matrix, but I didn't really get anywhere when trying this. Any tips?






coding-theory




coding-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 8 at 18:41









xzeoxzeo

388111




388111












  • $begingroup$
    $d+ d/3 leq n$ looks weird, as it would be simpler to write $ frac43 d leq n$ Are you sure you got it right?
    $endgroup$
    – leonbloy
    Jan 8 at 20:47












  • $begingroup$
    Yes I'm sure...
    $endgroup$
    – xzeo
    Jan 9 at 11:08










  • $begingroup$
    Hint: consider the possible columns of $G$ and how many linear combinatios produce a zero in each position....
    $endgroup$
    – leonbloy
    Jan 9 at 13:43






  • 2




    $begingroup$
    This is a special case of the Griesmer bound. The 2-dimensional case can, indeed, be proven the way leonbloy indicated. But the general result isn't very difficult either.
    $endgroup$
    – Jyrki Lahtonen
    Jan 9 at 15:25


















  • $begingroup$
    $d+ d/3 leq n$ looks weird, as it would be simpler to write $ frac43 d leq n$ Are you sure you got it right?
    $endgroup$
    – leonbloy
    Jan 8 at 20:47












  • $begingroup$
    Yes I'm sure...
    $endgroup$
    – xzeo
    Jan 9 at 11:08










  • $begingroup$
    Hint: consider the possible columns of $G$ and how many linear combinatios produce a zero in each position....
    $endgroup$
    – leonbloy
    Jan 9 at 13:43






  • 2




    $begingroup$
    This is a special case of the Griesmer bound. The 2-dimensional case can, indeed, be proven the way leonbloy indicated. But the general result isn't very difficult either.
    $endgroup$
    – Jyrki Lahtonen
    Jan 9 at 15:25
















$begingroup$
$d+ d/3 leq n$ looks weird, as it would be simpler to write $ frac43 d leq n$ Are you sure you got it right?
$endgroup$
– leonbloy
Jan 8 at 20:47






$begingroup$
$d+ d/3 leq n$ looks weird, as it would be simpler to write $ frac43 d leq n$ Are you sure you got it right?
$endgroup$
– leonbloy
Jan 8 at 20:47














$begingroup$
Yes I'm sure...
$endgroup$
– xzeo
Jan 9 at 11:08




$begingroup$
Yes I'm sure...
$endgroup$
– xzeo
Jan 9 at 11:08












$begingroup$
Hint: consider the possible columns of $G$ and how many linear combinatios produce a zero in each position....
$endgroup$
– leonbloy
Jan 9 at 13:43




$begingroup$
Hint: consider the possible columns of $G$ and how many linear combinatios produce a zero in each position....
$endgroup$
– leonbloy
Jan 9 at 13:43




2




2




$begingroup$
This is a special case of the Griesmer bound. The 2-dimensional case can, indeed, be proven the way leonbloy indicated. But the general result isn't very difficult either.
$endgroup$
– Jyrki Lahtonen
Jan 9 at 15:25




$begingroup$
This is a special case of the Griesmer bound. The 2-dimensional case can, indeed, be proven the way leonbloy indicated. But the general result isn't very difficult either.
$endgroup$
– Jyrki Lahtonen
Jan 9 at 15:25










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