Hamming Code Chinese
of 15
description
Hamming Code material
Transcript of Hamming Code Chinese
-
, ErrorCorrecting Codes,
ECC ,
ECC , ,
RAM ECC RAM,
RAM
?
, ,
,
, ,
(magnetic tape)
(magnetic disk); (com-
pact disk, CD) ,
, (CD
player) ,
(model) (1)
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data channelnoisy
data
1
(data)
(channel),
, CD ,
, ,
, ,
(noisy data),
,
.
? 1948
, ,
,
, ,
1948 Claude
Shannon A mathemat-
ical theory of communications,
,
,
, (channel capac-
ity), ,
, (code)
1
-
2 8312
, ,
,
, ,
(information theory),
ECC 1948
40,
Shannon
, Shannon ?
,
(en-
coder)(2)
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data Encoder Channeldataredundancy
noisy datanoisy redundancy
Decoder
gooddata
2
,
, (redun-
dant bits),
,
,
, (decoder),
,
.SingleError Correct-
ing(SEC) Codes
,
(sets) ABC (3.1)
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A B
C
5 2 61
3 4
7
3.1
1, 2, , 7,
1101, 1, 2, 3, 4
, bits ,
1, 3.2
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A B
C
0 1 11
0 1
0
3.2
bits 010,
1101010, ,
,
, 1101010,
6 bit , 10, 3.3a
-
3
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0 1 01
0 1
0
3.3a
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0 1 11
1 1
0
3.3b
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0 1 10
0 1
0
3.3c
,
1, A 1,
, C 1, , B
31, , 6
, 01,
, 3 bit,
, 3.3b, A 31,
, B41, , C 31,
, A C BC,
3, ,
bit, 3.3c,
,ABC,
1
, , ,
, single
error correcting (SEC)
(7, 4) Hamming code,
(7, 4)? 7, 4 bits
, (7, 4)
Hamming code? Ham-
ming 1950
, ? ,
67 bits , 3.4
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A B
C
0 1 01
0 1
1
3.1
A , B C ,
4 bit,
, , ,
bits ,
, , ?
, ,
bits ,
-
4 8312
3 bits,
, bits
,
,
1. (7, 4) Hamming Code
(7, 4)Hamming code
, ?
(binary arith-
metics), ,
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1 1 0 1 0 1
0 0 1 0 0 0
0 1 0 1
1
modulo 2,
2
,
, :x1 + x2 + x3 + x5 =0
x1 + x2 + x4 + x6 =0
x1 + x3 + x4 + x7 =0
xi {0, 1}
,
(linear equations),
codewords ?
H =
1 1 1 0 1 0 0
1 1 0 1 0 1 0
1 0 1 1 0 0 1
,
C codewords ,
x C xrow vector,
HxT = 0, CH null
space, dimension 7
rank 3, 4, C
24 = 16 codewords
, 4 data bits,
7 bits, 4 bits
, 2416, 16
codewords.
1.1 Syndrome
codewords
H null space, ?
codeword x,
e, 4
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...............Channelx y = x+ e
4
bit,ebit
1, 0, 1 1,
1, 10, e
syndrome, syndrome
-
5
, ,
, ,
syndrome ?
HyT , y = x + e,
,HxT +HeT , xcodeword,
HxT = 0, HeT , syn-
drome , code-
word , syndrome e
, , e,
codeword ,
24 = 16,
? 1,
,
, ,
,
, ,
yx , y =
(1101000) = (1101010) + (0000010),
syndrome
sT = HyT
=
1 1 1 0 1 0 0
1 1 0 1 0 1 0
1 0 1 1 0 0 1
1
1
0
1
0
0
0
=
0
1
0
,
e? e
bit 1, H
0
1
0
, e(0000010),
, 15,
, 1,
, (1101010)
codeword
?
, syndrome
0
0
0
,
bit ,
1
1
1
,
bit
1
1
0
, , bit
0
0
1
, syndromes ,
syndrome
, ,
, , (7,
4) Hamming code
1.2. Hamming Distance &
Hamming Weight
syndrome ,
Hamming distance
Hamming weight
Hamming distance?
x = (0001011) x
=
(1101010), Hamming distance
, dH(x, x
) =
-
6 8312
3, 3 bits Ham-
ming weight, 0
, wH(x) = 3, wH(x
) =
4 x x
Hamming
distance x + x
Hamming
weight, 0 + 0 = 0, 1 + 1 = 0,
1,
0 bits ,
bits , Hamming dis-
tance ,
minimum distance,
codewords Ham-
ming distance,
minimum distance:
dmin(C) = minx,x
Cx 6=x
dH(x, x
).
minimum weight,
0 codewords, Hamming
weight minimum
weight:
wmin(C) = minxCx6=0
wH(x).
minimum distance minimum
weight ,
(linear) , null space
linear subspace, ,
cordwords codeword,
codewords Hamming distance
Hamming weight,
, dmin(C) = wmin(C)
,
H =
1 1 1 0 1 0 0
1 1 0 1 0 1 0
1 0 1 1 0 0 1
null space
minimum distance ?
minimum distance minimum
weight, codeword
0, 1
1? ,
codeword weight 1, H
cloumn, 0,
codeword null space ,
1 2,
weight 2 codeword H,
H column , H
column ,
0, codeword
null space 3?
, weight3 codeword H
3 columns , col-
umn column
0
0
1
,
0
0
1
column, codeword
(1100001), weight 3,
codewords 7
, dmin = wmin = 3
, t,
? minimum distance
, t, minimum
distance 2t+1, ? 5
-
7
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t tx x
5
, codeword,
codeword Hamming
distance t (binary
vectors), t,
, , x
t, x
t,
x, x
,
t, 1,
, xx
t + 1 + t =
2t + 1, codewords
2t + 1, minimum distance
2t + 1 , min-
imum distance 2t + 1,
t,
,
codeword ,
t,
codewords,
2t + 1 minimum
distance 3, ,
,
2. SEC Hamming
Codes
(7, 4) Hamming code
, min-
imum distance 3, H
columns ,
columns , columns
, weight 2
codeword, H, columns
, 0, column
, columns
, ? rows
bits, columns
bits data , ,
rows , columns ,
, ,
3 rows, columns 7,
8? 8,
column ,
0
0
0
,
0
0
0
, ,
, syndrome
0
0
0
, , bit
,, 3 rows,
23 1 = 7 columns
? rows m,
2m 1 columns,
, :
H =
1 0 1
0 1 1... 0 1...
......
0 0 1
2m1
m
2m 1 bits,
H rows (linearly inde-
-
8 8312
pendent), H null space,
dimension 2m 1 rank m,
(2m1, 2mm1) SEC Hm-
maing code
.Double-Error Cor-
recting (DEC) Codes
,
,
,
, ,
, , ,
, ?
1. SEC DEC
?
, m =
4,
H=
0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
4 15, column
, ,
(15, 11) SEC Hamming code, 4 bits
, 11 bits data,
4 redundant bits ,
4 bits
, 4 rows,
,
, ,
, H
HSEC = (12 15), where 1
=
0
0
0
1
, 2 =
0
0
1
0
, , etc.
,
, 4 rows,
4 1 column vectors, i
,
HDEC =
1 2 3 15
1 2 3 15
i,
syndrome,
,
0
0...
0
, ,
column, ,
columns , ,
syndromes
, column columns
, i
i, i = f(i),
,
, , f
, ,
, , i
, ?
2. Finite Fields
-
9
?
0
0
0
1
0
0
1
0
? ,
, 4 1
columns16 binary vec-
tors, 16
field , field
, ,
, ,
field field
, , field
, ,
, ,
Abelian group,
,
, ;
, 0, Abelian
group; ,
, field
16 field, 16
0, 1, 2, , 9, A,B, C,D,E, F,
2,
, ,
Abelian group, 0
Abelian group, ,
field,?
Hexadecimal field
+ 0 1 2 3 4 5 6 7 8 9 A B C D E F
0 0 1 2 3 4 5 6 7 8 9 A B C D E F
1 1 0 3 2 5 4 7 6 9 8 B A D C F E
2 2 3 0 1 6 7 4 5 A B 8 9 E F C D
3 3 2 1 0 7 6 5 4 B A 9 8 F E D C
4 4 5 6 7 0 1 2 3 C D E F 8 9 A B
5 5 4 7 6 1 0 3 2 D C F E 9 8 B A
6 6 7 4 5 2 3 0 1 E F C D A B 8 9
7 7 6 5 4 3 2 1 0 F E D C B A 9 8
8 8 9 A B C D E F 0 1 2 3 4 5 6 7
9 9 8 B A D C F E 1 0 3 2 5 4 7 6
A A B 8 9 E F C D 2 3 0 1 6 7 4 5
B B A 9 8 F E D C 3 2 1 0 7 6 5 4
C C D E F 8 9 A B 4 5 6 7 0 1 2 3
D D C F E 9 8 B A 5 4 7 6 1 0 3 2
E E F C D A B 8 9 6 7 4 5 2 3 0 1
F F E D C B A 9 8 7 6 5 4 3 2 1 0
0 1 2 3 4 5 6 7 8 9 A B C D E F
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 A B C D E F
2 0 2 4 6 8 A C E 3 1 7 5 B 9 F D
3 0 3 6 5 C F A 9 B 8 D E 7 4 1 2
4 0 4 8 C 3 7 B F 6 2 E A 5 1 D 9
5 0 5 A F 7 2 D 8 E B 4 1 9 C 3 6
6 0 6 C A B D 7 1 5 3 9 F E 8 2 4
7 0 7 E 9 F 8 1 6 D A 3 4 2 5 C B
8 0 8 3 B 6 E 5 D C 4 F 7 A 2 9 1
9 0 9 1 8 2 B 3 A 4 D 5 C 6 F 7 E
A 0 A 7 D E 4 9 3 F 5 8 2 1 B 6 C
B 0 B 5 E A 1 F 4 7 C 2 9 D 6 8 3
C 0 C B 7 5 9 E 2 A 6 1 D F 3 4 8
D 0 D 9 4 1 C 8 5 2 F B 6 3 E A 7
E 0 E F 1 D 3 2 C 9 7 6 8 4 A B 5
F 0 F D 2 9 6 4 B 1 E C 3 8 7 5 A
2
p, Zp = {0, 1, 2, ,
p 1}, arithmetics modulo p
field field
? , p
-
10 8312
, 1, , p 1,
? Euclid
s algorithm,
, finite field,
field,
?
10 field? , finite
field ,
10, 10
field, 16 = 24 ,
4 = 22, ,
Z4, 4,
field, ,
2, modulo 4
, 1,
,4
field, , 16
, Z16, field,
, ?
, finite field
, pm
finite field, GF (pm),
Galois, fi-
nite field ,
20
pm finite field,
, Zp,
0p1, m,
m 1, m,
0p 1, p, pm,
,
Zp, modulo
a(x), a(x)m irreducible
polynomial, irreducible polynomial
,
, modulo a(x)
field , Zp
, irreducible polyno-
mial
, 216
field, a(x) = x4 + x + 1, bi-
nary polynomial ,
x4 +1, (x2+1)2,
x4+x2+x2+1, x2+x2 = 0,
x4+1,
,
x4 + x + 1,
irreducible polynomial
20, 1, 2, , F binary , 0
0000, 10001, 20010, 3
0011, , F 1111, poly-
nomials 0 0, 1 1, 2 x, 3
x + 1, , F x3 + x2 + x + 1,
, A1010, C1100, A+C = 0110,
6, 2A+ C = 6
, A C =? Ax3 + x, C
x3+x2, A C = x6+x5+x4+x3,
3, modulo
x4 + x + 1, 1,
2, A C = 1,
pm finite fields
, pm fields
field isomorphic,
, ,
-
11
isomorphic , finite field
, GF (pm)
,
finite field ,
, ,
, fi-
nite field multiplier,
,
(basis)
, ,
,
3. BCH Codes
, HDEC
columns
0
0
0
1
,
1
1
1
1
,
1, 2, 3, , F , columns
, ,
C3 = (C C)C = F C = 8,
,
HDEC=
(1 2 3 4 5 6 7 8 9 A B C D E F
1 8 F C A 1 1 A F F C 8 A 8 C
),
(15, 7),15 col-
umns, 8 rows, 8
, dimension= 15
8 = 7, 7 bits data, 8
bits , (15, 7)
BCH code, BCH
, BC code, Bose
RayChaudhuri 1960
, 1959
Hocquenghem,
, BCH code
minimum distance
5, ,
? , ,
H, codeword
0, He,
8 1, 4 1
4 1GF (16)
s1s3, sT = HeT =
(s1
s3
),
syndrome ,
, xy bits, sT =
(x
x3
)+(
y
y3
)=
(x+ y
x3 + y3
)=
(s1
s3
), ,
x + y = s1x3 + y3 = s3,
, s31 = (x + y)3 = x3 +
x2y + xy2 + y3, s3, s31 + s3 =
x2y + xy2 = xy(x + y), s1,
s13+s3s1
= xy, s1s3,
x + y xy, 1 = x +
y, 2 = xy, x+ yxyxy
, ,
,
:
2 + 1 + 2 = 0,
xy ,
?,
-
12 8312
x, y, s3 = s31, s
31 + s3 = 0,
2 = 0, 0,
, ,
,
, syndrome sT =(8
3
)=
(s1
s3
), 1 = s1 = 8, 2 =
s13+s3s1
= 83+3
8= A+3
8= 9
8= 9 81 =
9 F = E, 2 + 8 + E = 0
,
2,?
b
b24ac2a
, , 2a
0, ,
Try and Error,
? field ,
16, ,
15, GF (28),
255,255
,
, 255 ,
= 1, 1 + 8 + E = 7,
, 2, 4 + 3 + E = 9, , 3,
5+B+E = 0, , 3,
8 ,
3 + 8 = B, 3 B,
2,3 bit11
bit,
DEC BCH code ,
, x + y
xy,
, ,
,
algorithm , decoding algo-
rithm, BerlekampMassey al-
gorithm, Shannon
MIT ,
Euclid
s algorithm
, ?
3, ?
? 5
, 7 , ,
, t, 2t1
, columns
,3, 5, ,
, BCH
code
.Reed-Solomon Codes
,
,
,
, Reed,
Solomon, ReedSolomon code
nonbinary code, 1960
Reed Solomon ,
MIT Lincoln Laboratory,
, ,
, (n, k) code,
minimum distance n k+1,
binary ,
, bit binary,
-
13
symbol finite field
, , m
bits(6) finite field GF (2m)
,
10010101 m bits
01010011 m bits
6
? (7)
GF (2m) ,
, ,
, finite field
? Reed
Solomon :
, ,
1, 2, ,
,
, ,
, ,
(8),,min-
imum distance 5,
minimum distance n k + 1,
n k + 1 = 5, n k = 4, n k
, ,
, 3, 4, 5, 6
, ,
, 9
, ,
,
, ,
10, ,
, ,
, ,
, ,
,
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3.61.2 2.2
6.6
14.4
25.6
40.2
10
20
30
40
13
data, m
bits, (11),
,
()(12),
, (13),
,
data ,
,
data, ,
, ,
1,
1, ,
, t, 2t
,
finite field, fi-
nite field , ,
, ,
. Compact Disk Digi-tal Audio System
Compact Disk
Digital Audio System,
(digital) , ,
, , 0
1, ,
(analog) disk
12,,
74, 1.5 106 bits/sec,
746 109 bits ,
,
-
15
, track ,
, 3.5, track
0.5 106,
,
CrossInterleaved Reed
Solomon Codes, CIRC,
ReedSolomon Codes, (32,
28), (28, 24), symbol
8 bits, (32, 28), re-
dundancy 4, , (28,
24) ,
,
cross interleaving
? 32 bits 28
bits , 2824 bits ,
2832 24
28= 3
4, 3
4,
CD 6 109 bits ,
2109 bits
4000 bits
,2.5mm track,
8mm track, (detect)
, ,
, , ,
,
, CD
8mm,
CD
, ,
,
,
1. E. R. Berlekamp, ed., Key Papers in the
Development of Coding Theory. New
York: IEEE Press, 1974.
2. R. E. Blahut, Theory and Practice of
Error-Control Codes. Reading, MA:
Addison-Wesley, 1983.
3. S. Lin and D. J. Costello, Jr., Er-
ror Control Coding, Fundamentals and
Applications. Englewood Cliffs, NJ:
Prentice-Hall, 1983.
4. R. J. McEliece, The Theory of Infor-
mation and Coding. Reading, MA:
Addison-Wesley, 1977.
5. R. J. McEliece, Finite Fields for
Computer Scientists and Engineers.
Boston, MA: Kluwer Academic Pub-
lishers, 1987.
![Thema: „Hamming-Codes“kreuzer/Lehrerfortbil...1 Einführung in die Kodierungstheorie (aus Quelle: [Q1]) Ein Code, dahinter steckt nichts anderes als eine Information, die geschickt](https://static.fdocument.pub/doc/165x107/5b373ea27f8b9aad388e13f7/thema-hamming-codes-kreuzerlehrerfortbil1-einfuehrung-in-die-kodierungstheorie.jpg)
