INTRODUCTION TO ALGEBRAIC CODING THEORY

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Chapter 31

• INTRODUCTION TO ALGEBRAIC CODING THEORY

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Motivation

• Suppose you wish to send a message to either
  execute or not a particular command 1 to
  execute, 0 to not. The message is to be
  transmitted thousands of miles away and is
  susceptible to interference (noise) that could
  alter the intended message. What could be done to
  improve a safe delivery?

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Simple apply redundancy

• Start with the idea of majority rule. So if you
  want to send the message of 0, turn it into
  10001. Whichever number appears the most is the
  intended one.
• This involves the idea of independent error
  (maximum-likelihood decoding)
• So if you send a sequence of length 500 and each
  digit has a probability of error q.01 (p.99).
  SO the chance that the message is sent error free
  equals (.99)500 or approx. .0066

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Three fold repetition scheme

• 1001 -gt 111000111111
• 1 -gt111
• Error can only occur if 001,010,100,000
• Q (error) (.01)(.01)(.99)(.01)(.99)(.01)(.99)(.
  01)(.01)(.01)(.01)(.01) .000298
• P(no error) (.99702)500 .86

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Goal

• To devise reliable, efficient, and reasonably
  easy to implement
• Three basic features 1.set of
  messages 2.method of encoding 3. method of
  decoding

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Example 1
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Example 1 cont.
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Definition Linear Code

• An (n , k) linear code over a field F is a
  k-dimensional subspace V of the vector space
• F F F F
• n copies
• Over F. The members of V are called the code
  words. When F is Z2 the code is called binary.
• The message consist of k digits and the
  redundancy of the remaining n - k

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Example 4

• The set 0000,0121, 0212, 1022, 1110, 1201, 2011,
  2102, 2220 is a (4,2) linear code over Z3
  called a ternary code.

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Definitions

• The Hamming distance between two vectors of a
  vector space is the number of components in which
  they differ.
• Denoted d(u,v).
• The Hamming weight of a vector is the number of
  nonzero components of the vector. Denoted wt(u).
• The Hamming weight of a linear code is the
  minimum weight of any nonzero vector in the code.
• Example
• s 0010111, t 0101011, v 1101101
• d(s,t) 4, d(s,v) 5, wt(s) 4, wt(v) 5

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Theorem

• For any vectors u,v, and w, d(u,v) d(u,w)
  d(w,v) and
• d(u,v) wt(u v).
• This theorem and the previous definitions give
  the
• following.
• Theorem
• If the Hamming weight of a linear code is at
  least 2t 1,
• then the code can correct any to or fewer errors.
• Alternatively, the same code can detect any 2t or
  fewer
• errors.
• Note A linear code with Hamming weight 2t 1
  can detect t
• errors or 2t errors, not both.

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Example

• Hamming (7,4) code revisited (Table on p523)
• Hamming weight is 3 2 1 1.
• This code can detect 1 error or 1 or 2 errors.

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CHOSING THE G MATRIX
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Parity-check matrix decoding

• For any received word w, compute wH.
• If wH is the zero vector, assume no error was
  made.
• If there exist one instance of a nonzero element
  sF and a row i of H s.t. wH is s times row i,
  assume the sent word was w (0s0) where s
  occurs in the ith component. If there is more
  than one such instance, do not decode.
• If wH does not fit into either category 2 or 3,
  at least two errors occurred dont decode.

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ExampleHamming (7,4) codeThe generator matrix
G is
1 0 0 0 1 1 0
0 1 0 0 1 0 1
0 0 1 0 1 1 1
0 0 0 1 0 1 1
The parity-check matrix H is
1 1 0
1 0 1
1 1 1
0 1 1
1 0 0
0 0 1 0 0 1
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• Example (cont)
• Consider received word v 0000110, vH 110
• 110 is 1st row of H and no other, so an error was
  made in the 1st position of v.
• So the received word is really 1000110

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Lemma Orthogonality Relation

• Let C be an (n, k) linear code over F with
  generator matrix G and parity-check matrix H.
  Then for any vector in Fn, we have vH 0 (the
  zero vector) iff v belongs to C

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Theorem 31.3 Parity-Check Matrix Decoding

• Parity-check matrix decoding will correct any
  single error
• if and only if
• the rows of the parity-check matrix are nonzero
  and no one row is a scalar multiple of any other.

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Coset Decoding

• Do this by constructing a table called a standard
  array.
• 1st row is the set C of received code words
  beginning with the identity 0 . . . 0
• choose another element v in V and consider the
  coset v C
• (where v is not an element already listed in
  the table)
• among the elements of v C, choose one of
  minimum weight,
• call it v.
• create the next row by placing vector v c
  under the code
• word c
• continue until all vectors in V have been
  listed
• (Note an (n, k) linear code over a field with
  q elements will have
• v c q( n- k) rows
• 1st column is called the coset leaders

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ExampleConsider (6,3) binary linear codeC
000000, 100110, 010101, 001011, 110011, 101101,
011110, 111000
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Theorem 31.4 Coset Decoding Is Nearest-Neighbor
Decoding

• In coset decoding, a received word w is decoded
  as a code word c such that d( w, c) is a minimum

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DefinitionIf an (n, k) linear code over F has a
parity-check matrix H, then, for any vector u in
Fn, the vector uH is called the syndrome of u.

• Theorem
• Let C be an (n, k,) linear code over F with a
  parity-check matrix
• H. Then, two vectors of F n are in the same
  coset of C iff they
• have the same syndrome.
• We can use syndromes for decoding any received
  word w
• Calculate wH (syndrome)
• Find coset leader v such that wH vH
• Assume vector sent was w - v

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Example 11
1 1 0
1 0 1
0 1 1
1 0 0
0 1 0
0 0 1

• H
• Coset leaders 000000 100000 010000 001000
  000100 000010 000001 100001
• Syndromes 000 110 101
  011 100 010 001 111
• For v 101001, vH 100. Since the coset leader
  000100 has 100 as its syndrome, v-000100 101101
  was sent.