Hamming codes

History
 In the late 1940’s Richard Hamming recognized
that the further evolution of computers required
greater reliability, in particular the ability to not
only detect errors, but correct them. His search
for error-correcting codes led to the Hamming
Codes, perfect 1-error correcting codes, and the
extended Hamming Codes, 1-error correcting
and 2-error detecting codes.

Uses
 Hamming Codes are still widely used in
computing, telecommunication, and other
applications.
 Hamming Codes also applied in
 Data compression
 Some solutions to the popular puzzle The Hat
Game
 Block Turbo Codes

A [7,4] binary Hamming Code
 Let our codeword be (x1 x2 … x7) ε F2
7
 x3, x5, x6, x7 are chosen according to the message
(perhaps the message itself is (x3 x5 x6 x7 )).
 x4 := x5 + x6 + x7 (mod 2)
 x2 := x3 + x6 + x7
 x1 := x3 + x5 + x7

[7,4] binary Hamming codewords

 Let a = x4 + x5 + x6 + x7 (=1 iff one of these
bits is in error)
 Let b = x2 + x3 + x6 + x7
 Let c = x1 + x3 + x5 + x7
 If there is an error (assuming at most one) then
abc will be binary representation of the
subscript of the offending bit.

 If (y1 y2 … y7) is received and abc ≠ 000, then
we assume the bit abc is in error and switch it.
If abc=000, we assume there were no errors (so
if there are three or more errors we may recover
the wrong codeword).

Definition: Generator and Check
Matrices
 For an [n, k] linear code, the generator matrix is
a k×n matrix for which the row space is the
given code.
 A check matrix for an [n, k] is a generator matrix
for the dual code. In other words, an (n-k)×k
matrix M for which Mx = 0 for all x in the
code.

A Construction for binary Hamming
Codes
 For a given r, form an r × 2r-1 matrix M, the
columns of which are the binary representations
(r bits long) of 1, …, 2r-1.
 The linear code for which this is the check
matrix is a [2r-1, 2r-1 – r] binary Hamming Code
= {x=(x1 x2 … x n) : MxT = 0}.

Example Check Matrix
 A check matrix for a [7,4] binary Hamming
Code:

Syndrome Decoding
 Let y = (y1 y2 … yn) be a received codeword.
 The syndrome of y is S:=LryT. If S=0 then
there was no error. If S ≠ 0 then S is the binary
representation of some integer 1 ≤ t ≤ n=2r-1
and the intended codeword is
x = (y1 … yr+1 … yn).

Example Using L3
 Suppose (1 0 1 0 0 1 0) is received.
100 is 4 in binary, so the intended codeword was
(1 0 1 1 0 1 0).

Extended [8,4] binary Hamm. Code
 As with the [7,4] binary Hamming Code:
 x3, x5, x6, x7 are chosen according to the message.
 x4 := x5 + x6 + x7
 x2 := x3 + x6 + x7
 x1 := x3 + x5 + x7
 Add a new bit x0 such that
 x0 = x1 + x2 + x3 + x4 + x5 + x6 + x7 . i.e., the new
bit makes the sum of all the bits zero. x0 is called a
parity check.

Extended binary Hamming Code
 The minimum distance between any two
codewords is now 4, so an extended Hamming
Code is a 1-error correcting and 2-error
detecting code.
 The general construction of a [2r, 2r-1 - r]
extended code from a [2r –1, 2r –1 – r] binary
Hamming Code is the same: add a parity check
bit.

Check Matrix Construction of
Extended Hamming Code
 The check matrix of an extended Hamming
Code can be constructed from the check matrix
of a Hamming code by adding a zero column on
the left and a row of 1’s to the bottom.

Decoding Extended Hamming Code

q-ary Hamming Codes
 The binary construction generalizes to
Hamming Codes over an alphabet A={0, …, q},
q ≥ 2.
 For a given r, form an r × (qr-1)/(q-1) matrix M
over A, any two columns of which are linearly
independent.
 M determines a [(qr-1)/(q-1), (qr-1)/(q-1) – r]
(= [n,k]) q-ary Hamming Code for which M is
the check matrix.

Example: ternary [4, 2]
Hamming
 Two check matrices for the some [4, 2] ternary
Hamming Codes:

Syndrome decoding: the q-ary case
 The syndrome of received word y, S:=MyT, will
be a multiple of one of the columns of M, say
S=αmi, α scalar, mi the ith column of M. Assume
an error vector of weight 1 was introduced y =
x + (0 … α … 0), α in the ith spot.

Example: q-ary Syndrome
 [4,2] ternary with check matrix ,
word (0 1 1 1) received.
 So decode (0 1 1 1) as
(0 1 1 1) – (0 0 2 0) = (0 1 2 1).

Perfect 1-error correcting
 Hamming Codes are perfect 1-error correcting
codes. That is, any received word with at most
one error will be decoded correctly and the code
has the smallest possible size of any code that
does this.
 For a given r, any perfect 1-error correcting
linear code of length n=2r-1 and dimension n-r
is a Hamming Code.

Proof: 1-error correcting
 A code will be 1-error correcting if
 spheres of radius 1 centered at codewords cover the
codespace, and
 if the minimum distance between any two
codewords ≥ 3, since then spheres of radius 1
centered at codewords will be disjoint.

Proof: 1-error correcting
 Suppose codewords x, y differ by 1 bit. Then
x-y is a codeword of weight 1, and M(x-y) ≠ 0.
Contradiction. If x, y differ by 2 bits, then
M(x-y) is the difference of two multiples of
columns of M. No two columns of M are
linearly dependent, so M(x-y) ≠ 0, another
contradiction. Thus the minimum distance is at
least 3.

Perfect
 A sphere of radius δ centered at x is
Sδ(x)={y in An : dH(x,y) ≤ δ}. Where A is the
alphabet, Fq, and dH is the Hamming distance.
 A sphere of radius e contains
words.
 If C is an e-error correcting code then
, so .

Perfect
 This last inequality is called the sphere packing
bound for an e-error correcting code C of
length n over Fm:
where n
is the length of the code and in this case e=1.
 A code for which equality holds is called perfect.

Proof: Perfect
 The right side of this, for e=1 is qn/(1+n(q-1)).
 The left side is qn-r where n= (qr-1)/(q-1).
qn-r(1+n(q-1)) = qn-r(1+(qr-1)) = qn.

Applications
 Data compression.
 Turbo Codes
 The Hat Game

Data Compression
 Hamming Codes can be used for a form of
lossy compression.
 If n=2r-1 for some r, then any n-tuple of bits x
is within distance at most 1 from a Hamming
codeword c. Let G be a generator matrix for
the Hamming Code, and mG=c.
 For compression, store x as m. For
decompression, decode m as c. This saves r bits
of space but corrupts (at most) 1 bit.

The Hat Game
 A group of n players enter a room whereupon
they each receive a hat. Each player can see
everyone else’s hat but not his own.
 The players must each simultaneously guess a
hat color, or pass.
 The group loses if any player guesses the wrong
hat color or if every player passes.
 Players are not necessarily anonymous, they can
be numbered.

The Hat Game
 Assignment of hats is assumed to be random.
 The players can meet beforehand to devise a
strategy.
 The goal is to devise the strategy that gives the
highest probability of winning.

Source
 “Notes on Coding Theory” by J.I. Hall
http://www.mth.msu.edu/~jhall/classes/coden
otes/coding-notes.html

Hamming codes

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Hamming codes