In error detection and correction, majority logic decoding is a method to decode repetition codes, based on the assumption that the largest number of occurrences of a symbol was the transmitted symbol.
In a binary alphabet made of
0
,
1
, if a
(
n
,
1
)
repetition code is used, then each input bit is mapped to the code word as a string of
n
-replicated input bits. Generally
n
=
2
t
+
1
, an odd number.
The repetition codes can detect up to
[
n
/
2
]
transmission errors. Decoding errors occur when the more than these transmission errors occur. Thus, assuming bit-transmission errors are independent, the probability of error for a repetition code is given by
P
e
=
∑
k
=
n
+
1
2
n
(
n
k
)
ϵ
k
(
1
−
ϵ
)
(
n
−
k
)
, where
ϵ
is the error over the transmission channel.
The code word is
(
n
,
1
)
, where
n
=
2
t
+
1
, an odd number.
Calculate the
d
H
Hamming weight of the repetition code.
if
d
H
≤
t
, decode code word to be all 0's
if
d
H
≥
t
+
1
, decode code word to be all 1's
In a
(
n
,
1
)
code, if R=[1 0 1 1 0], then it would be decoded as,
n
=
5
,
t
=
2
,
d
H
=
3
, so R'=[1 1 1 1 1]
Hence the transmitted message bit was 1.
