SME
Communication Systems · Problem type
Block code rate and correction
Use this for “The (7, 4) Hamming code has minimum distance 3. Find its rate and the errors it corrects and detects.”
This is an example. Type your own values to start a new problem.
R
=
k
n
t
=
⌊
−
1
2
⌋
e
=
d
−
1
Formulas
R
=
k
n
0.5714
=
4
7
t
=
⌊
d
− 1
2
⌋
1
=
⌊
3 − 1
2
⌋
e
=
d
−
1
2 = 3 − 1
Assumptions
•
An (n, k) block code sends each k message bits as an n-bit codeword.
•
The minimum distance d is the fewest bits in which two codewords differ.
•
Up to t errors leave a word nearest its own codeword; up to d − 1 cannot make another codeword.
Step-by-step
WE KNOW
n = 7 bit, k = 4 bit, d = 3
FIND
code rate (R), errors corrected (t), errors detected (e)
Step 1 · Find code rate (R)
Use R = k ÷ n
The share of each codeword that is message; the rest is checking.
R
=
4
7
R
=
0.5714
Step 2 · Find errors corrected (t)
Use t = ⌊(d − 1) ÷ 2⌋
A
word
with
t
errors
must
stay
nearer
its
own
codeword
than
any
other,
so
2
t
<
d
.
t
=
⌊
3 − 1
2
⌋
t
=
1
Step 3 · Find errors detected (e)
Use e = d − 1
Fewer
than
d
flips
can
never
turn
one
codeword
into
another,
so
they
are
seen.
e
=
3
−
1
e
=
2
Check
Put the numbers back in. Both sides should match.
0.5714
=
4
7
✓
1
=
⌊
3 − 1
2
⌋
✓
2 = 3 − 1 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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