SJ ¨ALVST ¨ANDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET

SJ ¨ ALVST ¨ ANDIGA ARBETEN I MATEMATIK

MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET

Line¨ ara koder och Reed-Muller koder

av

Magnus Weiderling

2012 - No 14

MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM

Line¨ ara koder och Reed-Muller koder

Magnus Weiderling

Sj¨ alvst¨ andigt arbete i matematik 15 h¨ ogskolepo¨ ang, grundniv˚ a Handledare: Rikard B¨ ogvad

2012

n

A ={s1, s2, . . . , sq}

q A

q n A = x₁x₂. . . x_n

xi 2 A i 1 i  n (x1, . . . , xn)

q n A C q

n

C C

C C |C|

M

R(C) C n ^logq_n^|C|

C M = qⁿ

q n n = logq M = log M/log q

C

q n

A =F2={0, 1}

0 1 q = 2

C1={00, 01, 10, 11} n = 2 |C| = 4 C2={000, 011, 101, 110} n = 3 |C| = 4

F3={0, 1, 2} F4

q

q Fq

q Fq

n A

= x₁. . . x_n = y₁. . . y_n ( ) =P_n

i=1d(xi, yi)

xi yi 1

(xiyi) =

⇢ 1 xi6= yi

0 xi= yi

C

C d(C)

(C) = {d( ) : 2 6= }

C1 = {00, 01, 10, 11}

x = 00 y = 11 x y

d(x, y) = 1 + 1 = 2 C1 d(C1) = 1

n M d (n, M, d)

n, M d

0)

C ={000, 010, 111} wt(010) = 1

q

q

p p < ¹₂

q 1

{0, 1}

(1 | 0 ) = (0 | 1 ) = p

C

p Mr

Mr c c2 C

Mc

Mr Mc

(Mr Mc = max Mr c 2 C

Mr Mc

d(Mr, Mc)

d(M_r, M_c= min{d(Mr, c) 2 C

p < ¹₂

C Mr

n n 0 i  n

d(Mr, c) = i, P Mr pⁱ(1 p)^{n i} p < ¹₂

p⁰(1 p)ⁿ> p¹(1 p)^{n 1}> ... > pⁿ(1 p)⁰

Mr

2 C Mr

d(M_r, c)

u C

u u

u

u (u + 1)

C ={00000, 00111, 11111}

1

C 1

2

00111 11111

C u

d(C) u + 1

d(C) u+1 2 C 1 d( ) u

2 C/ u d(C) 1 < d(C) C u

d(C) < u + 1 d(C)  u 1 2 2 C

1 d 1 2) = d(C) u 1

d(C) 2 C

C u

v C

v v C

v v (v + 1)

C ={000, 111}

000 100, 010

001 000

111 110, 101

011 111

C 1

C v d(C)

2v + 1

d b(d 1)/2c

( d(C) 2v + 1

v d( ) v

02 C 6= ⁰

d( ⁰, ) d( ⁰, ) d( , )

d( ⁰ ) d( 2v + 1 v = v + 1 > d(

C v

) C v d(C) < 2v + 1

0 2 C d( ⁰ = d(C) 2v v

C v d(C) 2v + 1

d( ⁰ < v + 1 ⁰

v

0 2 C C v

d( ⁰ v+1 ⁰

d = d(C) v + 1 d  2v

x1. . . xv

| {z }

lika som c⁰

xv+1. . . xd

| {z }

lika som c

xd+1. . . xn

| {z }

lika som c⁰ och c

d( ⁰ ) = d v  v = d( d( ⁰ ) <

d( d( ⁰ = d(

0

V Fq

{e1, . . . , er} V

a1e1+ ... + arer= 0) a1= ... = ar= 0

S V Fq S ={e1, e2, . . . , ek}

V < S >

< S > ={(a1e1+· · · + ake_k) : ai2 Fq}

S < S > {0}

q = 2 S = {0001, 0010, 0100}

< S > ={0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111}

V Fq

S ={e1, . . . , ek} V V V = < S > S

V Fq

V Fq

dim _Fq(V )

= (v1, . . . , vn)

= (w1, . . . , wn)2 Fⁿq

· v1·w1+...+vn·wn2 Fq

· = 0 Fⁿ_q Fⁿ_q

n Fq S^? S

S^? { 2 Fⁿq : · = 0 2 S}

S S^?=Fⁿq

n Fq

C n Fq

Fⁿ_q

C n k Fq

[n, k] [n, k, d] d

M k (n, M ) = (n, q^k) C

(e1, ..., e_k)

(a1e1+ ... + akek) ai Fq q

C q^k

k = logqq^k= logqM

C1={000000, 100000, 010000, 110000}

C2={000000, 111000, 000111, 111111}

C1 q = 2 4 = 2² 6 k = 2

1, 1 2 1

C2 k = 2 3, 3 6

3

3 1

C Fⁿq C C^?

C Fⁿq

d(C) C

C Fq d(C) = wt(C) wt(C)

C d( = wt(

d(C) ^{0 0} 2 C

d( ^{0 0}) = d(C)

d(C) = d( ^{0 0}) = wt( ^{0 0} wt(C) ^{0 0} 2 C

2 C { } wt(C) = wt(

wt(C) = wt( ) = d( d(C)

S

Fⁿq S

S

C

q = 3 C = < S >

S ={12101, 20110, 01122, 11010}

0 BB

@

1 2 1 0 1 2 0 1 1 0 0 1 1 2 2 1 1 0 1 0

1 CC A !

0 BB

@

1 2 1 0 1 0 2 2 1 1 0 1 1 2 2 0 2 2 1 2

1 CC A !

0 BB

@

1 2 1 0 1 0 1 1 2 2 0 0 0 0 1 0 0 0 0 0

1 CC A

C G

C G C

C [n, k] k⇥ n

k n

n k

k k 1

k + 1

H

H C

C^?

C [n, k] (n k)⇥

n

C u C uH^T = 0

m C

G H

(I_k|X) I_k

k⇥k X G

|

(Y|In k)

Y H

H

G C

G C

C [n, k] Fq G

2 Fⁿq C^?

G 2 C^?, G^T =

(n k)⇥ n H H

C H HG^T =

i i G i2 C 1 i  k

2 C

= _{1 1}+ ... + _{k k 1}, . . . , _k2 Fq

2 C^? · = 0 2 C i

1 i  k G^T =

· i= 0 1 i  k = 1 1+...+ k k 2

C

· = 1( · 1) +· · · + k( · k) = 0

H C

H H C^?

HG^T =

HG^T = H

C^? H

(n k) C^?

H C

G = (I_k|X)

C H = ( X^T|In k) C

HG^T = 0

n k H

X^T = X^T X

C H

C

C d d 1 H

C  d H d

= (v1, ..., vn)2 C e > 0 i₁, ..., i_e v_j = 0 j

2 {i/ 1, ..., ie} i 1  i  n i H

C = (v1, ..., vn) e

vi1, ..., vie

= H^T v_i1 ^T_i1 v_ie ^T_ie

e H ci1, ..., cie

C

d C

 d 1  d 1

H (i)

C

d C

 d H

 d (ii)

C

H = 0

@ 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1

1 A

H 0

d(C) = 3

C [n, k, d] Fq

C C q^k

( 1, ..., k)

= u1 1+ ... + uk k u1, ..., uk2 Fq

G C i

i = (u₁, ..., u_k)2 F^kq

= G = u1 1+ ... + uk k C

2 C = G = (u₁, ..., u_k)2 F^kq

2 F^kq = G

C (5, 3)

G_(5,3)= 0

@ 1 0 1 1 0 0 1 0 1 1 0 0 1 0 1

1 A

= 101

= G = (101) 0

@ 1 0 1 1 0 0 1 0 1 1 0 0 1 0 1

1 A =

= 1 + 0 + 0, 0 + 0 + 0, 1 + 0 + 1, 1 + 0 + 0, 0 + 0 + 1 = 1, 0, 0, 1, 1

1 3

C n

Fⁿq 2 Fⁿq n

C

C+ ={( + ) : 2 C} (= + C) q = 2 C ={000, 101, 010, 111}

C + 000 ={000, 101, 010, 111}

C + 001 ={001, 100, 011, 110}

C + 010 ={010, 111, 000, 101}

C + 011 ={011, 110, 001, 100}

C + 100 ={100, 001, 110, 011}

C + 101 ={101, 000, 111, 010}

C + 110 ={110, 011, 100, 001}

C + 111 ={111, 010, 101, 000}

C + 000 = C + 010 = C + 101 = C + 111 = C C + 001 = C + 011 = C + 100 = C + 110 =F³2 C

C 2 C

= 2

+C = 2 C

+C =

q = 2 C ={0000, 1011, 0101, 1110}

w = 1101 0101

C + 0000 ={0000, 1011, 0101, 1110}

C + 0001 ={0001, 1010, 0100, 1111} (= C + 0100) C + 0010 ={0010, 1001, 0111, 1100}

C + 1000 ={1000, 0011, 1101, 0110}

w = 1101

u = 1000 w u = 1101 1000 =

1101 + 1000 = 0101

C + 0000

n n

C [n, k, d] Fq

H C 2 Fqⁿ

) = H^T 2 Fq^{n k}

H

C C =

{0000, 1011, 0101, 1110}

G =

✓ 1 0 1 1 0 1 0 1

◆

S( ) = H^T

0000, 0001, 0010 1000

C HG^T =

H =

✓ 1 0 1 0 1 1 0 1

◆

S( )

S( )

q = 2 C = {0000, 1011, 0101, 1110}

= 1101 S( ) = H^T = 11

1000 = 1101 1000 = 0101

n, M d n M d

q C

(n, M, d) C (C) = (d 1)/n

d/n

R(C) = ^log_n^qM

C ={00 . . . 0| {z }

n

, 11 . . . 1| {z }

n

} Fq

(n, M, d) = (n, 2, n) [n, k, d] = [n, 1, n]

M = 2

k = dim(C) = logq(M ) = log2(2) = 1

log2(2)

n = _n¹ ! 0 n! 1 (C) = ^{n 1}_n ! 1 n! 1

R(C)

A q (q > 1)

n d Aq

M (n, M, d) A

A_q(n, d) = max{M : (n, M, d) A}

(n, M, d) C M = Aq

q^k q n

d Bq(n, d) q^k

[n, k, d] Fq

Bq(n, d) = max{q^k: [n, k, d] Fq}

Aq(n, d) Bq(n, d) Aq(n, d)

q = 2 C C

C Fq C

C ={(c1, ..., cn, P_n

i=1(ci)) : (c1, ...cn)2 C}

q = 2 Pn

i=1(ci) =Pn i=1(ci)

C ={000, 111, 011, 100}

[3, 2, 1] C

P3

i=1(ci) = 0 + 0 + 0 = 0 P3

i=1(ci) = 1 + 1 + 1 = 1 P₃

i=1(ci) = 0 + 1 + 1 = 0 P3

i=1(ci) = 1 + 0 + 0 = 1

C ={0000, 1111, 0110, 1001} [4, 2, 2]

Aq(n, d)

q > 1 n, d 1 d  n

qⁿ Pd 1

i=0(ⁿi)^{(q 1)i}  Aq

A_q(n, d) B_q(n, d)

A q, q > 1

2 Aⁿ A n r 0

r

SA( , r) ={ 2 Aⁿ: d( ) r}

Aⁿ

q > 1 n r 0 V_qⁿ(r)

V_qⁿ(r) =

⇢ _n

0 + ⁿ₁ (q 1) + ⁿ₂ (q 1)²+ ... + ⁿ_r (q 1)^r 0 r  n

qⁿ n r

r 0 Aⁿ r

V_qⁿ(r) A q > 1

2 Aⁿ 2 Aⁿ

d( , ) = m Aⁿ m

m

n

m q 1

m _mⁿ (q 1)^m

0 r  n

n  r SA( , r) = Aⁿ V_qⁿ(r) = qⁿ

q > 1 n d 1 d  n

Aq(n, d) P_{b(d 1)/2c}^qn i=0 (ⁿi)^{(q 1)i}

b(d 1)/2c  (d 1)/2

C = { 1 2 M} (n, M, d)

A A q M = Aq(n, d)

e = b(d 1)/2c SA( i, e) Aⁿ

S_M

i=1SA( i, e)✓ Aⁿ

|Aⁿ| = qⁿ

|SA( _i, e)| = Vqⁿ(e) i

M · Vqⁿ(e) qⁿ =) Aq(n, d) = M _V^qnⁿ

q(e) = _Vn ^qn q(b(d 1)/2c)

q

qⁿ P_{b(d 1)/2c}

i=0 (ⁿi)^{(q 1)i}

r 2

n = 2^r 1 H

F^r2 2^r 1 Ham(r, 2)

Ham(3, 2)

[7, 4, 3] 7 4

7 3

47

G G

G_Ham(3,2)= 0 BB

@

1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1

1 CC A

H_Ham(3,2)= 0

@ 1 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0 0 1

1 A

H_Ham(3,2) G_Ham(3,2) I4

H_Ham(3,2) I7 4 = I3

G_Ham(3,2) H_Ham(3,2)

(i)

(ii) Ham(r, 2) k = 2^r 1 r

(iii) Ham(r, 2) 3 1

(iv) (i)

(ii) H Ham(r, 2) 2^r 1 r⇥

(2^r 1) 2^r 1 r

(iii) H

H H

1 0 . 0

0 1 . 0

1 1 . 0

Ham(r, 2)

3 1

(iv)

Ham(r, 2)

|C| = Pb(d 1)/2c^qn

i=0 (ⁿi)^{(q 1)i} = _1+n²ⁿ q = 2 d = 3 n = 2^r 1

|C| = ₁₊₂^22rr1₁ = 2^{2r 1 r}

2^k (ii)

Ham(r, 2) 1 2^r

2^r (= n + 1) n

 1 n + 1 j

j

j jH^T j

H H

( ) = H^T ) = 0

6= 0 j 1 j  2^r 1

j

j j

Ham(3, 2)

= 1001001

H^T = 010 H

2= 0100000 + 2= 1101001

Ham(r, 2) Ham(r, 2)

Ham(r, 2)

2^r 1 r

2^r Ham(r, 2) ^2r₂r^{1 r}1 ) Ham(r, 2)

(n, M, d) C Fq

R(C) = ^log_n^qM (C) =^{d 1}_n

M n

d Aq(n, d) q, n

d A_q(n, d)

= x1...xn = y1...yn

[ = (x1. . . xny1. . . yn)

C_i [n, k_i, d_i] Fq

i = 1, 2 C ={( + ) : 2 C1, 2 C2}

[2n, k1+ k2, min{2d1, d2}] Fq

D [n, k, d] C

C ={( ) : 2 D} [ {( , + ) : 2 D}

[2n, k + 1, min{n, 2d}]

C1= D C2={ , }

0, 8 0, 2

0, 8 log2(1/0, 8) + 0, 2

log2(1/0, 2)⇡ 0, 722

V = 0

S = 1 L1 = 0, 2·

1 + 0, 8· 1 = 1 0, 722 N

0, 722N 2

V V = 0 V S = 10 SV = 110 SS = 111 L2 = 0, 64· 1 + 0, 16· 2 + 0, 16 · 3 + 0, 04 · 3 = 1, 56

N/2 L2N/2 = 0, 78N

3 8

= 2, 184 0, 728N 0, 722

RM (r, m)

RM (1, 5)

RM (1, 5) 2⁵= 32

16 7 15

r m

0  r  m r RM (r, m)

[2^m, ^m₀ + ^m₁ + ... + ^m_r , 2^{m r}] r = 1

r > 1

r = 0

(RM (0, m))

0 RM (0, m) m 0

{0...0, 1...1} 2^m

0 2^m

(RM (1, m), )

RM (1, m)

m 1

(i) RM (1, 1) =F²2={00, 01, 10, 11}

(ii) m + 1 2 RM (1, m + 1) =

{( ) : 2 RM(1, m)} [ {( + ) : 2 RM(1, m)}

(RM (1, 2) )

(ii) = 00

{00, 00} [ {00, 11} 0000 0011

RM (1, 1) 8 RM (1, 2)

0000 0011 0101 0110 1001 1010 1100 1111

[4, 3, 2]

RM (1, 2)

G_{RM (1,2)}= 0

@ 1 1 1 1 0 1 0 1 0 0 1 1

1 A

(RM (1, 3) ) RM (1, 3) 16

00000000 00001111 00110011 00111100 01011010 01010101 01100110 01101001 10010110 10011001 10100101 10101010 11000011 11001100 11110000 11111111 [8, 4, 4]

4

m 1 RM (1, m) [2^m, m + 1, 2^{m 1}]

0 1 2^{m 1}

RM (1, 1) [2, 2, 1]

RM (1, m) RM (1, m 1)

RM (1, m 1)

2^{m 1}, m, 2^{m 2}] RM (1, m)

2· 2^{m 1}, m + 1, min{2 · 2^{m 2}, 2^{m 1}}] = [2^m, m + 1, 2^{m 1}]

RM (1, m + 1)

2^{(m+1) 1} = 2^m RM (1, m + 1)

RM (1, m) 2 RM(1, m)

2^{m 1} 2^{m 1}+ 2^{m 1}= 2· 2^{m 1}= 2^m

2 RM(1, m)

2^{m 1} = ²₂^m RM

2^{m 1} 2^m

= 0 = 1 2^m

= 1 = 0 2^m

RM (1, 2)

2^{2 1} = 2 1

RM (1, 3) 2^{3 1}= 4 1

RM (1, m+

1) RM (1, m)

(i) RM (1, 1) G_{RM (1,1)}

✓ 1 1 0 1

◆

(ii) G_{RM (1,m)} RM (1, m)

RM (1, m + 1) G_{RM (1,m+1)}=

✓ G_{RM (1,m)} G_{RM (1,m)} 0...0 1...1

◆

G_{RM (1,2)}

RM (1, 3)

G_{RM (1,3)} 0 BB

@ 1 x1

x2

x3

1 CC A

0 BB

@

1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1

1 CC A

RM (1, 3)

k =

✓ m 0

◆ +

✓ m 1

◆ +

✓ m 2

◆ +· · · +

✓ m r

◆

= 4 r = 1 m = 3

4 Muc= (a0a1a2a3)

a_i Mc =P3

i=0aiG_{RM (1,3)i} G_{RM (1,3)i} i

Muc= 1010 RM (1, 3)

Mc= 1· (11111111) + 0 · (00001111) + 1 · (00110011) + 0 · (01010101) =

= 11111111 + 00110011 = 11001100 Mc

Muc

1

RM (1, 3)

= x1. . . xn = y1. . . yn

x y

\ = (x1· y1. . . xn· yn)

\ 1 i 1 i

x1, . . . , xm

xi1xi2. . . xis xi x1, . . . , xm

xixj = xjxi

x²_i = xi

p(x1, . . . , xm) Z2

0

1 m = 3 2·2·2 = 8

p(x1, x2, x3) = 1· x1+ 1· x2+ 0· x3

x1 x2 x3 p

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 0

1 1 1 0

p

ap= 00111100 RM

00000000 11111111

(RM (r, m))

0  r  m r

ap n = 2^m

p(x1, ..., xm) r

RM (1, 3) r = 1 m = 3 n = 2³= 8

{1, x1, x2, x3} 1 x1, x2

x3

p(x1, x2, x3) 1

p(x1, x2, x3) = a01 + a1x1+ a2x2+ a3x3

ai= 0 1

a0(11111111) + a1(00001111) + a2(00110011) + a3(01010101)

16 a0 = 0 a1= 1 a2= 1 a3= 0

x1+ x2 00001111 + 00110011 = 00111100

k

S k Z^m2 m

S b + S b2 Z^m2 dim(S) = k b + S

k

S

H S

x2 S () xH^T = 0

x2 b + S () x b2 S (x b)H^T = 0

x2 b + S xH^T = bH^T

Z^m2

EG(m, 2) m Z2

dim(S) = 0 b + S 0

b+{0} = {b} EG(m, 2)

EG(m, 2) 1

{b, b + c} c2 Z^m₂ {0}

Z²2 EG(2, 2)

2²= 4 0 ⁴₂ = 6 1

EG(3, 2) 2³= 8 0 ⁸₂ = 28 1

8

3 / ⁴₃ = 14 2

8 EG(3, 2)

ap=

EG(3, 2) ap ap

1 F = 2

RM (1, m) m 3

k = m 1

m = 1, k = 0 RM (1, 1) = Z¹₂ EG(1, 2)

2¹= 2 ²₂ = 1

0 RM (1, 1)

m = 2 RM (1, 2) = Z²2

EG(2, 2) (2²) = 4 ⁴₂ = 6

1 RM (1, 2)

m = 3 RM (1, 3) 14

2

F E EG(m, 2) aF · aE ⌘ |F \ E|

mod 2

|F | F = 0

0 1 aF · aE ⌘

wt(aF \ aE) = wt(aF\E) =|F \ E|

F = + S E = + T EG(m, 2)

Z^m₂ F \ E = ; F \ E = + (S \ T )

F \ E

F \ E = ; F \ E 6= ; x

F \ E x + r2 F \ E r2 S \ T

y F\ E x + r r2 S \ T

x = b + s x2 b + S r2 S \ T

S x + r = b + (s + r) 2 b + S

s + r2 S x + r2 c + T

x + r b + S c + T F \ E

y F \ E x = b + s

y = b + s1 x y S x y2 T

x y2 T \ S x y r y = x + r

0

RM (r, m)

p(x₁, . . . , x_m) = Xr s=0

X

i1,...,is

a_i1,...,isx_i1. . . x_is= a₀+ a₁x₁+· · · + a12x₁x₂+ . . .

r

ap= Xr s=0

X

i1,...,is

ai1,...,isax_i1...x_is

a_xi1...xis x_i1· · ·

xis ax1x2= x1· x2

p(x1, . . . , xm)

ap

r > 1 r, r 1, r 2

RM (1, 3) r = 1, m = 3

ap= a01 + Xm i=1

aiaxi

{k}^c={j1, . . . , jm 1}

{1, . . . , m} {1, 2, 3}

RM (1, 3) {k} = {3} {k}^c={1, 2}

a_p· ax_j1···x_{jm 1}

ap

a_xi· ax_j1...x_{jm 1} ⌘ |Fxi\ Fx_j1...x_{jm 1}| mod 2 ⌘ |Fxix_j1...x_{jm 1}| mod 2

Fx1 x1

0 Fxix_j1...x_{jm 1} 0

{i, j1, . . . , jm 1} = {1, . . . , m}

{i} = {j1. . . jm 1}^c={k}

RM (1, 3) {k} = {3}

{3} = {1, 2}^c={3}

ap· ax_j1...x_j2 = ai

ap· ab+F_{xj1 ...xj2}

b + F_xj1...xj2 F_xj1...xj2

axi· ab+F_{xj1 ...xj2} ⌘ |Fxi\ (b + Fx_j1...x_j2)| mod 2

|Fxi\ (b + Fx_j1...x_j2)| = 1 2 Fxi\ (b + Fx_j1...x_j2)

Fxi\ (b + Fx_j1...x_j2) = + (Fxi\ Fx_j1...x_j2) = + Fxix_j1...x_j2

0 Fxix_j1...x_{jm 1}

ap· ab+Fxj1 ...xjm 1 = ai

{j1. . . jm 1} = {k}^c b EG(3, 2)

2^{m 1} 1

Fx_j1...x_j2

2^{m 1} 4 RM (1, 3) ai

(2^{m 1} 1)/2

RM (1, 3)

1 x1 x2 x3 x1· x2 x2· x3 x1· x3

1 0 0 0 0 0 0

1 0 0 1 0 0 0

1 0 1 0 0 0 0

1 0 1 1 0 1 0

1 1 0 0 0 0 0

1 1 0 1 0 0 1

1 1 1 0 1 0 0

1 1 1 1 1 1 1

RM (1, m)

1 2^{m 1}

a0

RM (1, 3) M_c= (c₀c₁...c₇)

Mr = (r0r1...r7) Me = (e0e1...e7)

Mr= Mc+ Me ap

Mc = 00111100

M_r= 10111100 Mc {010, 011, 100, 101}

Mc= (a0a1a2a3)G_{RM (1,3)}= a01 + a1x1+ a2x2+ a3x3

a0, a1, a2 a3

a_x1x2 F_x1x2 ={110, 111}

ax2x3 Fx2x3 ={011, 111}

ax1x3 Fx1x3 ={101, 111}

a3 k = 3 {k}^c={1, 2}

Fx1x2

{110, 111}

{100, 101}

{010, 011}

{000, 001}

a3= ap· ab+F_{xj1 ···xj2}

10111100· 00000011 = 0 10111100· 00001100 = 0 10111100· 00110000 = 0 10111100· 11000000 = 1

a3 0

a2= a1= 1 a0

Mr a1a2a3· GRM (1,3)2,3,4= 10111100 110· GRM (1,3)2,3,4

= 10111100 00111100 = 10000000 = a0· 1 + Me

a0 1 01111111 a0 0

10000000 a0= 0

0 1

0, 1, 1 0

Muc= 0110 Me

Mr= 10111100 Mr Me= Mr+ Me= 00111100

RM (0, 1) 00 11

RM (1, 1) 00 01 10 11

RM (1, 2) 0000 0011 0101 0110 1001 1010 1100 1111

RM (1, 3)

00000000 00001111 00110011 00111100 01011010 01010101 01100110 01101001 10010110 10011001 10100101 10101010 11000011 11001100 11110000 11111111

RM (1, 4)

0000000000000000 0000000011111111 0000111100001111 0000111111110000 0011001111001100 0011001100110011 0011110000111100 0011110011000011 0110011010011001 0101101010100101 0110100110010110 0101010110101010 0101010101010101 0101101001011010 0110100101101001 0110011001100110 1111000000001111 1111000011110000 1100110011001100 1001011010010110 1100001100111100 1100001111000011 1100110000110011 1010101010101010 1001100110011001 1010010110100101 1010101001010101 1001100101100110 1010010101011010 1001011001101001 1111111100000000 1111111111111111

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M =|C| C

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Mc

Mr

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n p

q A

R(C) C

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si i A

r_i i

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wt(x) x

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