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Error Detection Codes & Modular Arithmetic in Information Theory

Explore the concepts of parity checks, weighted codes, modular arithmetic, and error probabilities in information theory. Learn how to apply these principles to detect and correct errors in data transmission systems.

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Error Detection Codes & Modular Arithmetic in Information Theory

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  1. Chapter 2 Parity checks Modular arithmetic Weighted codes

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. ISBN Numbers 10 digit weighted code (mod 11) where X stands for ten in the check (last) digit 2.9

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