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7/28/2019 Gallian Ch 31
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7/28/2019 Gallian Ch 31
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A k-dimensional subspace Vof the vector space
...nF F F F (n copies) over a field F. Themembers ofVare called code words. When Fis
2Z ,
the code is called binary.
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Hamming distance
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The number of components in which two vectors of avector space differ. Notation: d(u,v).
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Hamming weight
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The number of non-zero components of a vectoru.Notation: wt(u).
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Hamming weight of a linear code
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The minimum weight of any nonzero vector in the code.
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Properties of Hamming Distance and Hamming Weight
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For any vectors u, v, and w, ( , ) ( , ) ( , )d u v d u w d w v
and ( , ) ( )d u v wt u v .
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Correcting Capabilities of a Linear Code
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If the Hamming weight of a linear code is at least 2 1t ,then the code can correct any tor fewer errors.
Alternatively, the code can detect any 2t or fewererrors.
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Standard generator matrix(or standard encoding matrix)
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A kby n matrix in which the first kcolumns are theidentity matrix. When using this transformation matrix,
the message constitutes the first kcomponents of thetransformed vectors.
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Systematic code
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An (n,k) linear code in which the kinformation digitsoccur at the beginning of each code word.
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Parity-check matrix
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Let [ | ]k
G I A be the standard generator matrix.
The ( )n n k matrix Hwhose first krows are A and
whose next n k rows are the identity matrix.
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Decoding procedure using parity-check matrix(steps 1-3 of 4)
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Step 1: For any received word w, compute wH.
Step 2: IfwHis the zero vector, assume no error wasmade.
Step 3: If there is exactly one instance of a nonzeroelement s F and a row IofHsuch that wHis s times
row i, assume the sent word was (0,..., ,...,0)w s ,
where s occurs in the i-th component. If there is morethan one such instance, do not decode.
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Decoding procedure using parity-check matrix(steps 3 and 4 of 4)
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Step 3: When the code is binary, category 3 reduces tothe following. IfwHis the i-th row ofHfor exactly one i,
assume that an error was made in the i-th component ofw. IfwHis more than one row ofH, do not decode.
Step 4: IfwHdoes not fit into either category 2 orcategory 3, we know that at least two errors occurred intransmission and we do not decode.
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Orthogonality Relation
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Let Cbe an (n,k) linear code overFwith generatormatrix G and parity-check matrix H. Then, for any
vectorvin nF , we have 0vH (the zero vector) if anyonly ifvbelongs to C.
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Parity-Check Matrix Decoding
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Parity-check matrix decoding will correct any singleerror if and only if the rows of the parity-check matrix
are nonzero and no one row is a scalar multiple of anyother.
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Standard array
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A table constructed as follows.
The first row is the set ofCcode words, beginning incolumn 1 with the identity 0...0 .
To form additional rows, choose an element vofVnotlisted so far. Among all the elements of the coset v C ,choose one of minimum weight, say v. Complete thenext row of the table by placing under the columnheaded by the code word cthe vector 'v c .
Continue this process until all vectors in Vhave beenlisted.
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Coset leader
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A word in the first column of the standard array.
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Coset Decoding Is Nearest-Neighbor Decoding
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In coset decoding, a received word wis decoded as a
code word csuch that ( , )d w c is a minimum.
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Syndrome ofu
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An (n,k) linear code overFhas parity matrix H.
For any vectorn
u F , the vectoruH.
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Same Coset---Same Syndrome
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Let Cbe an (n,k) linear code overFwith parity-checkmatrix H. Then, two vectors of nF are in the same
coset ofCif and only if they have the same syndrome.
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Procedure for decoding a word in coset decoding
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1. Calculate wH, the syndrome ofw.2. Find the coset leadervsuch that wH vH .
3. Assume that the vector sent was w v .