Download Simple Block Code Parity Checks

Chapter 2
Parity checks
Modular arithmetic
Weighted codes
Simple Block Code Parity Checks
Idea: Break a string of bits into blocks of size (n  1) and
append (or prepend) an additional bit for even or odd
parity (even # of 1’s, odd # of 1’s), to obtain n bit blocks.
n 1
parity
bit
b0 , b1 ,..., bn 1



n-1 bits of
informatio n
Set b0   bi (mod 2), so that
i 1
n 1
b
i 0
i
2 0
n
1
Bit Redundancy 
 1 
 excess redundancy
n 1
n 1
Other codes that admit parity checks: 2-out-of-5, 3-out-of-7 (van Duuren code)
In general,
Redundancy 
2length


log 2 (# of possible blocks)
log 2 (2 n )
n


log 2 (# of allowable (valid)) log 2 (2 n 1 ) n  1



# of symbols
2.4
Error Probabilities
Assume independent error probability p for each bit.
The probability of no error = (1 – p)n.
The probability of one error = np(1 – p)n-1.
i
In general
1
√ … √
n


n
i
n i


1  (1  p )  p      p (1  p )
i 
i 0 

n
exactly i errors
n
√ … √
Burst Code Parity Checks
Assume noise comes in “bursts” of length ≤ L,
and is otherwise independently distributed.
Instead of computing a parity check over
contiguous sequences of bit positions, use a
“checksum” over words of length L.
physical
noise
b1
…
bL
bL+1
…
b2L
.
.
.

c1
bits



n 1
So, instead of b1 bn 1bn with bn   bi (mod 2)
…
cL
checksum
i 1
words


n 1
we do w1  wn 1wn with word wn   wi (mod 2)
i 1
where wi  L, and the sum is logical addition.
2.6
Modular Arithmetic
In mod (modulo) 2 arithmetic, 2 is the base (modulus) and there are no
numbers other than 0 and 1. Any higher number mod 2 is obtained by
dividing it by 2 and taking the remainder. For instance, 3 ≡ 1 mod 2 and
4 ≡ 0 mod 2.
Mod 2 multiplication
Mod 2 addition
×
+ 0 1
0 1
0
0
1
1
1
0
= logical XOR
0
0
0
1
0
1
= logical AND
The base can be any number (usually a prime) and the rules are similar to
those for base 2.
Mod 5 addition
Mod 5 multiplication
+
0
1
2
3
4
×
0
1
2
3
4
0
0
1
2
3
4
0
0
0
0
0
0
1
1
2
3
4
0
1
0
1
2
3
4
2
2
3
4
0
1
2
0
2
4
1
3
3
3
4
0
1
2
3
0
3
1
4
2
4
4
0
1
2
3
4
0
4
3
2
1
2.8
Rules for Modular arithmetic
Fact: If a ≡m a′ and b ≡m b′, then a + b ≡m a′ + b′ and a ∙ b ≡m a′ ∙
b′.
Proof: a ≡m a′ and b ≡m b′ means that a = a′ + im and b = b′ + jm,
so a + b = a′ + b′ + (i + j)m and a ∙ b = a′ ∙ b′ + a′jm + b′im + ijm2.
QED
Multiplicative inverses may not exist for some numbers.
Example: 2 × 5 ≡ 0 mod 10. Does 2 have a multiplicative
inverse? Suppose it does, then 2 × 2−1 ≡ 1 mod 10. However,
multiplying both sides by 5 yields 0 ≡ 5 mod 10, which is false.
Note: If the modulus is a prime, p, then numbers not congruent to
p can’t multiply together to be congruent to p, so you don’t have
the problem in the above example. Here is a proof that
multiplicative inverses (for nonzero) numbers always exist in a
prime modulus, p: Let q  , 0 < q < p, then GCD(p, q) = 1. (They
are relatively prime). By the Euclidean Algorithm, we can always
2.8
find k, l   such that kp + lq = 1,  lq ≡ 1 mod p  l = q−1.
Weighted Codes
Deals with transcription errors (i.e. human non-binary error):
Example: Assign numerical values 1 … 37 to the symbols
consisting of letters, digits, and a space. Take a message of
length less than 37 and weight each symbol therein according
to its position in the message, appending a final check symbol
s1 chosen so the weighted sum is congruent to zero:
n
For n < 37,
sn… … s1
 ks
k 1
k
 0 (mod 37)
transposing adjacent digits [e.g. 67 → 76]: ksk + (k+1)sk+1
becomes (k+1)sk + ksk+1 whose difference is sk+1 − sk ≢ 0 (mod 37)
repeating an adjacent digit [e.g. 667 → 677]: difference is
ksk − ksk+1 = k(sk − sk+1) ≢ 0 provided 0 < k < 37 and sk ≠ sk+1
ISBN Numbers
10 digit weighted code (mod 11) where X stands for ten in the check (last) digit
10
9
8
7
6
5
4
3
2
1
0  1 3 2 1  2 5 7 1  4
message
highest weight:
w
sum
→
w
sum of sums
→
↓
x
→
w+x
↓
→
↓
y
→
w+x+y
z
→
w+x+y+z
2w + x
↓
→
↓
check digit (weight 1):
w
3w + 2x + y
↓
→
4w + 3x + 2y + z
2.9