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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. parity bit.

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chapter 2

Chapter 2

Parity checks

Modular arithmetic

Weighted codes

simple block code parity checks
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.

parity

bit

Other codes that admit parity checks: 2-out-of-5, 3-out-of-7 (van Duuren code)

In general,

2.4

error probabilities
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.

In general

burst code parity checks

Assume noise comes in “bursts” of length ≤ L, and is otherwise independently distributed.

physical

noise

Burst Code Parity Checks

checksum

Instead of computing a parity check over contiguous sequences of bit positions, use a “checksum” over words of length L.

2.6

modular arithmetic

Mod 2 multiplication

Mod 2 addition

Modular Arithmetic

= logical AND

= logical XOR

The base can be any number (usually a prime) and the rules are similar to those for base 2.

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 5 addition

Mod 5 multiplication

2.8

rules for modular arithmetic
Rules for Modular arithmetic

Fact: If a≡ma′ and b ≡mb′, then a + b ≡ma′ + b′ and a ∙ b ≡ma′ ∙ b′.

Proof: a≡ma′ and b ≡mb′ 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 find k, l  such that kp + lq = 1,  lq≡ 1 mod p  l = q−1.

2.8

weighted codes
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:

For n < 37,

sn… … s1

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
ISBN Numbers

10 digit weighted code (mod 11) where X stands for ten in the check (last) digit

2.9