What are linear codes having minimum distance 2 used for?
$begingroup$
Consider the following parity check matrix
$$H = begin{bmatrix}
1 & 0 & 1 & 1 & 1 & 1 \
0 & 1 & 1 & 0 & 0 & 1 \
1 & 1 & 0 & 1 & 0 & 1
end{bmatrix}$$
Since its 1st and 4th column both are identical, the minimum distance of a code is 2. If we compute the syndrome of any received vector $vec{r}$ and it comes out to be zero, we can't possibly tell whether the received vector is correct or not (as 1st and 4th bit of $vec{r}$ could be incorrect).
If we can't verify whether the received vector is correct or not, what are linear codes having minimum distance used for?
linear-algebra matrices coding-theory parity
asked Dec 5 '18 at 23:51
HeisenbergHeisenberg
1183
$endgroup$
add a comment |
$begingroup$
Consider the following parity check matrix
$$H = begin{bmatrix}
1 & 0 & 1 & 1 & 1 & 1 \
0 & 1 & 1 & 0 & 0 & 1 \
1 & 1 & 0 & 1 & 0 & 1
end{bmatrix}$$
Since its 1st and 4th column both are identical, the minimum distance of a code is 2. If we compute the syndrome of any received vector $vec{r}$ and it comes out to be zero, we can't possibly tell whether the received vector is correct or not (as 1st and 4th bit of $vec{r}$ could be incorrect).
If we can't verify whether the received vector is correct or not, what are linear codes having minimum distance used for?
linear-algebra matrices coding-theory parity
asked Dec 5 '18 at 23:51
HeisenbergHeisenberg
1183
$endgroup$
$begingroup$
Really no matter what distance your code has, you can't possible tell whether the received vector is correct or not if there is more than 1 code word (in a distance $d$ code, you could potentially have $d$ errors). So what are you really asking here?
$endgroup$
– Morgan Rodgers
Dec 8 '18 at 3:42
add a comment |
$begingroup$
Consider the following parity check matrix
$$H = begin{bmatrix}
1 & 0 & 1 & 1 & 1 & 1 \
0 & 1 & 1 & 0 & 0 & 1 \
1 & 1 & 0 & 1 & 0 & 1
end{bmatrix}$$
Since its 1st and 4th column both are identical, the minimum distance of a code is 2. If we compute the syndrome of any received vector $vec{r}$ and it comes out to be zero, we can't possibly tell whether the received vector is correct or not (as 1st and 4th bit of $vec{r}$ could be incorrect).
If we can't verify whether the received vector is correct or not, what are linear codes having minimum distance used for?
linear-algebra matrices coding-theory parity
asked Dec 5 '18 at 23:51
HeisenbergHeisenberg
1183
$endgroup$
Consider the following parity check matrix
$$H = begin{bmatrix}
1 & 0 & 1 & 1 & 1 & 1 \
0 & 1 & 1 & 0 & 0 & 1 \
1 & 1 & 0 & 1 & 0 & 1
end{bmatrix}$$
Since its 1st and 4th column both are identical, the minimum distance of a code is 2. If we compute the syndrome of any received vector $vec{r}$ and it comes out to be zero, we can't possibly tell whether the received vector is correct or not (as 1st and 4th bit of $vec{r}$ could be incorrect).
If we can't verify whether the received vector is correct or not, what are linear codes having minimum distance used for?
linear-algebra matrices coding-theory parity
linear-algebra matrices coding-theory parity
asked Dec 5 '18 at 23:51
HeisenbergHeisenberg
1183
asked Dec 5 '18 at 23:51
HeisenbergHeisenberg
1183
edited Dec 6 '18 at 1:27
Heisenberg
asked Dec 5 '18 at 23:51
HeisenbergHeisenberg
1183
asked Dec 5 '18 at 23:51
HeisenbergHeisenberg
1183
asked Dec 5 '18 at 23:51
HeisenbergHeisenberg
1183
1183
$begingroup$
Really no matter what distance your code has, you can't possible tell whether the received vector is correct or not if there is more than 1 code word (in a distance $d$ code, you could potentially have $d$ errors). So what are you really asking here?
$endgroup$
– Morgan Rodgers
Dec 8 '18 at 3:42
add a comment |
$begingroup$
Really no matter what distance your code has, you can't possible tell whether the received vector is correct or not if there is more than 1 code word (in a distance $d$ code, you could potentially have $d$ errors). So what are you really asking here?
$endgroup$
– Morgan Rodgers
Dec 8 '18 at 3:42
$begingroup$
Really no matter what distance your code has, you can't possible tell whether the received vector is correct or not if there is more than 1 code word (in a distance $d$ code, you could potentially have $d$ errors). So what are you really asking here?
$endgroup$
– Morgan Rodgers
Dec 8 '18 at 3:42
$begingroup$
Really no matter what distance your code has, you can't possible tell whether the received vector is correct or not if there is more than 1 code word (in a distance $d$ code, you could potentially have $d$ errors). So what are you really asking here?
$endgroup$
– Morgan Rodgers
Dec 8 '18 at 3:42
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
If the minimum distance is $d=2t+1$ with $tgeq 1$, then the code can correct $t$ errors. The minimum value for $d$ is 3. A code with minimum distance 2 can only detect one error but not correct it.
answered Dec 6 '18 at 6:42
WuestenfuxWuestenfux
4,2531413
$endgroup$
add a comment |
$begingroup$
The typical (practically, the only) linear code with minimum distance 2 is the single parity code which consist of adding a single parity bit, hence $n=k+1$. It detects a single error. See some uses in the page above.
Given that your code is $(6,3)$, it looks rather bad, because for $k=3$ you could have the single parity $(4,3)$ with better rate and same minimum distance. It's true that minimum distance is not the only parameter to judge the quality of the code (more important is the probability of erroneous decoding), but it looks bad anyway.
answered Dec 8 '18 at 3:37
leonbloyleonbloy
40.6k645107
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If the minimum distance is $d=2t+1$ with $tgeq 1$, then the code can correct $t$ errors. The minimum value for $d$ is 3. A code with minimum distance 2 can only detect one error but not correct it.
answered Dec 6 '18 at 6:42
WuestenfuxWuestenfux
4,2531413
$endgroup$
add a comment |
$begingroup$
If the minimum distance is $d=2t+1$ with $tgeq 1$, then the code can correct $t$ errors. The minimum value for $d$ is 3. A code with minimum distance 2 can only detect one error but not correct it.
answered Dec 6 '18 at 6:42
WuestenfuxWuestenfux
4,2531413
$endgroup$
add a comment |
$begingroup$
If the minimum distance is $d=2t+1$ with $tgeq 1$, then the code can correct $t$ errors. The minimum value for $d$ is 3. A code with minimum distance 2 can only detect one error but not correct it.
answered Dec 6 '18 at 6:42
WuestenfuxWuestenfux
4,2531413
$endgroup$
If the minimum distance is $d=2t+1$ with $tgeq 1$, then the code can correct $t$ errors. The minimum value for $d$ is 3. A code with minimum distance 2 can only detect one error but not correct it.
answered Dec 6 '18 at 6:42
WuestenfuxWuestenfux
4,2531413
answered Dec 6 '18 at 6:42
WuestenfuxWuestenfux
4,2531413
answered Dec 6 '18 at 6:42
WuestenfuxWuestenfux
4,2531413
answered Dec 6 '18 at 6:42
WuestenfuxWuestenfux
4,2531413
4,2531413
add a comment |
add a comment |
$begingroup$
The typical (practically, the only) linear code with minimum distance 2 is the single parity code which consist of adding a single parity bit, hence $n=k+1$. It detects a single error. See some uses in the page above.
Given that your code is $(6,3)$, it looks rather bad, because for $k=3$ you could have the single parity $(4,3)$ with better rate and same minimum distance. It's true that minimum distance is not the only parameter to judge the quality of the code (more important is the probability of erroneous decoding), but it looks bad anyway.
answered Dec 8 '18 at 3:37
leonbloyleonbloy
40.6k645107
$endgroup$
add a comment |
$begingroup$
The typical (practically, the only) linear code with minimum distance 2 is the single parity code which consist of adding a single parity bit, hence $n=k+1$. It detects a single error. See some uses in the page above.
Given that your code is $(6,3)$, it looks rather bad, because for $k=3$ you could have the single parity $(4,3)$ with better rate and same minimum distance. It's true that minimum distance is not the only parameter to judge the quality of the code (more important is the probability of erroneous decoding), but it looks bad anyway.
answered Dec 8 '18 at 3:37
leonbloyleonbloy
40.6k645107
$endgroup$
add a comment |
$begingroup$
The typical (practically, the only) linear code with minimum distance 2 is the single parity code which consist of adding a single parity bit, hence $n=k+1$. It detects a single error. See some uses in the page above.
Given that your code is $(6,3)$, it looks rather bad, because for $k=3$ you could have the single parity $(4,3)$ with better rate and same minimum distance. It's true that minimum distance is not the only parameter to judge the quality of the code (more important is the probability of erroneous decoding), but it looks bad anyway.
answered Dec 8 '18 at 3:37
leonbloyleonbloy
40.6k645107
$endgroup$
The typical (practically, the only) linear code with minimum distance 2 is the single parity code which consist of adding a single parity bit, hence $n=k+1$. It detects a single error. See some uses in the page above.
Given that your code is $(6,3)$, it looks rather bad, because for $k=3$ you could have the single parity $(4,3)$ with better rate and same minimum distance. It's true that minimum distance is not the only parameter to judge the quality of the code (more important is the probability of erroneous decoding), but it looks bad anyway.
answered Dec 8 '18 at 3:37
leonbloyleonbloy
40.6k645107
answered Dec 8 '18 at 3:37
leonbloyleonbloy
40.6k645107
answered Dec 8 '18 at 3:37
leonbloyleonbloy
40.6k645107
answered Dec 8 '18 at 3:37
leonbloyleonbloy
40.6k645107
40.6k645107
add a comment |
add a comment |
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$begingroup$
Really no matter what distance your code has, you can't possible tell whether the received vector is correct or not if there is more than 1 code word (in a distance $d$ code, you could potentially have $d$ errors). So what are you really asking here?
$endgroup$
– Morgan Rodgers
Dec 8 '18 at 3:42