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• 2 Key system parameters available for designer
– Ch BW
– Transmitted Sig Power
• Hence we go for modulation schemes but error
performance is an issue – hence error control coding
• Another motivation is to reduce Eb/ N0 for a fixed BER
to reduce Tx power to reduce hardware cost
IMPLICATIONS OF CH CAP THEOREM

• Block Codes
Divide the message into blocks, each of k bits,
called datawords. Then add r redundant bit to
make length n=k+r
The resulting n-bit blocks are called
codewords.
• Convolution Codes
Absence of memory (nutshell- serial input no
blocks)
ERROR CORRECTION CODES

DEFINITIONS
• WORD:- Sequence of symbol.
• CODE :- Set of vectors called the codeword.
• HAMMING WEIGHT :- No of non-zero elements.
• HAMMING DISTANCE :- No of places the codeword differs.
D(c1, c2) = w(c1-c2)
• BLOCK LENGTH :- Length of code words in a BLOCK CODE,
denoted by “n”.
• MSG BITS :- “k”
• CODE RATE :- Of an (n,k) code , the ratio of (k/n), and reflects
the fraction of the codeword that consists the info.

• MINIMUM DISTANCE :- Minimum hamming
distance b/w any two codes.
• MINIMUM WEIGHT(w) :- The smallest weight of
any nonzero codeword.
DEFINITIONS

LINEAR CODES
• Properties
– Addition of two codewords gives third codeword.
– All zero code is also a codeword.
– Minimum distance = minimum weight.

LINEAR CODES
K- msg bits (m0, m1, m2, m3,…………… mk-1) b-parity bits (b0, b1, b2,
b3,…………… bn-k-1)
Codeword(c0, c1, c2, c3,…………… cn-1)
n – code length
Ci = bi (parity bit) i= 0,1,2,………n-k-1.
mi+k-n(msg bits) i= n-k, n-k+1, ……..n-1

• The (n-k) parity bits are linear sum of k msg bits.
• bi = p0i m0 + p1i m1 + ………….+ pk-1i mk-1
• Where the coeff are defined as follows:-
pij = 1 if bi depends on mj
0 otherwise
LINEAR CODES

Coefficient matrix P
p00 p01 …….. P0n-k-1
p10 p11 …….. P1,n-k-1
P = . . .
. . .
pk-1,0 pk-1,1 ……. Pk-1,n-k-1
Where pi = 0 or 1
Identity Matrix : Ik =
1 0 0 0…. 0
0 1 0 0…. 0
0 0 1 0…. 0
. . . .
. . . .
0 0 0 1 ….1

 All code words can be obtained as linear combination of
basis vectors.
 The basis vectors can be designated as {𝑔1, 𝑔2, 𝑔3,….., 𝑔 𝑘}
 For a linear code, there exists a k by n generator matrix such
that
𝑐1∗𝑛 = 𝑚1∗𝑘 . 𝐺 𝑘∗𝑛
where c={𝑐1, 𝑐2, ….., 𝑐 𝑛} and m={𝑚1, 𝑚2, ……., 𝑚 𝑘}
GENERATOR MATRIX

G = [ 𝐼 𝑘 P]
C = m.G = [m mP]
Message
part Parity part
m=(1110) and G = =
c= m.G = 𝒎 𝟏 𝒈 𝟏 + 𝒎 𝟐 𝒈 𝟐 + 𝒎 𝟑 𝒈 𝟑 + 𝒎 𝟒 𝒈 𝟒
= 1. 𝒈 𝟏 + 𝟏. 𝒈 𝟐 + 𝟏. 𝒈 𝟑 + 𝟎. 𝒈 𝟒
c = (1101000) + (0110100) + (1110010)
= (0101110)
1 1 0 1 0 0 0
0 1 1 0 1 0 0
1 1 1 0 0 1 0
1 0 1 0 0 0 1
𝒈 𝟎
𝒈 𝟏
𝒈 𝟐
𝒈 𝟑
Example:
Let us consider (7, 4) linear code where k=4 and n=7

• For decoding we require to separate the parity
bits and also find out if there was an error.
• We use a parity check matrix H s.t
H = [ PT I n-k ]
And H GT = 0
c HT = 0
Thus m c
c 0 null vector
GENERATOR
MATRIX
PARITY
CHECK
MATRIX T

CYCLIC CODES
• They form a sub class of linear block codes
• Properties
– Linearity a+b = c
– Cyclic property ie any cyclic shift of a code word is
also a code word.
BCH CODES ARE LINEAR CYCLIC BLOCK CODES

CHARACTERISTICS OF BCH CODES
• For positive pair of integers m≥3 and t, a (n, k) BCH
code has parameters:
– Block length: n = 2m – 1
– Number of check bits: n – k ≤ mt
– Minimum distance: dmin ≥ 2t + 1
• t<(2m – 1)/2 random errors detected and corrected.
• So also called ‘t-error correcting BCH code’.
• Major advantage is flexibility for block length and code
rate.
K- msg bits (m0, m1, m2, m3,…………… mk-1)
b-parity bits (b0, b1, b2,
b3,…………… bn-k-1)
n – code length

MANIFESTATION OF CHARACTERISTICS
• The parameters of some useful BCH codes are:
n = 2m – 1
n
n – k ≤ mt
k
t<(2m – 1)/2
t
Generator Polynomial
7 4 1 1 011
15 11 1 10 011
15 7 2 111 010 001
15 5 3 10 100 110 111
31 26 1 100 101
31 21 2 11 101 101 001
31 16 3 1 000 111 110 101 111
31 11 5 101 100 010 011 011 010 101
31 6 7 11 001 011 011 110 101 000 100
111

• Generator polynomial g(x)  specified in
terms of its roots from Galois Field GF(2k).
• g(x) has α,α2,…, α2t and their conjugates as its
roots.
• We choose g(x) from xn + 1 polynomial factors
by taking xn-k as highest term.
CHARACTERISTICS OF BCH CODES
Galois
Field
Root Elements Min Poly
Genr Poly

GALOIS FIELD
• Evariste Galois (1832)
Field +, -,
x, /
Ring
+, -
x
Gp
+, -

AXIOMS FOR FINITE FIELDS
1. Field F is closed under + and x iea+b and a.b are in F if a & b are in
F
2. Associative law : (a+b)+c = a+(b+c), a.(b.c) = (a.b).c
3. Commutative law : a+b=b+a, a.b=b.a
4. Distributive : a.(b+c)= a.b+a.c
5. Identity elements 1 & 0 must exist in F s.t
6. a+0 =a
7. a.1 = a
8. For any a there is additive inverse s.t a+(-a)=0
9. For any a except 0 there is multiplicative inverse (a= 𝑎−1
) s.t
a(𝑎−1) =1
10. A field with finite elements is called Galois Field denoted by
GF(q).
11. If 1st 8 properties are satisfied then its called a ring.

STRUCTURE OF FINITE FIELDS
• Gaolis Field : Fq ={0, α, α2, …, αq-1}.
• Primitive element : An element α in a finite field
Fq is called a primitive element (or generator) of
Fq if Fq ={0, α, α2, …, αq-1}.
For q = pm
• Where : q represents the number of elements in
the field.
p is a prime number
α is the primitive element
ord[α] = q-1 and αq-1 = 1
m is degree of primitive polynomial
• Now, π(α) is a primitive polynomial ϵ Fp [α] = 0
of degree m

• Example : Consider the field GF(5)=(0,1,2,3,4) , we will check if 2
and 3 are primitive elements
• 2 𝑜 = 1 (mod 5) = 1 3 𝑜 = 1 (mod 5) = 1
• 21 = 2 (mod 5) = 2 31 = 3 (mod 5) = 3
• 22 = 4 (mod 5) = 4 32 = 9 (mod 5) = 4
• 23 = 8 (mod 5) = 3 33 = 27 (mod 5) = 3

MODULAR ARITHMETIC
• Use only limited range of integers.
• We, define upper limit, called a ,modulus N.
• Then use only the integers 0 to N-1.
• This is modulo-N arithmetic.

Modulo-2 Arithmetic
Here modulus N is 2. we can use only 0
and 1. operation in this arithmetic are
very simple.
The addition and subtraction give the
same results. Here, we use XOR operation
for both the add and sub.
The result of an XOR operation is 0(if both
the bits are same. The result is 1 if the
any of the two bit is different.)

MODULO OPS
Mod 5 addition Mod 5 multiplication
for GF(5)

MIN POLY
• If βϵFqit is a root of xq -1 which is a poly with coeff in Fp[x]
• Min poly of β is : Mβ[x] = poly of least degree in Fp[x]
s.t Mβ[β] = 0 ie……
• Min poly is always monic.
• Mβ(x) = π(x− 𝜸)
𝜸𝛜conjugates of β = β,βp , βp2 ………..
Example of GF(16)
• Generator poly g(x)

• Message D: (0110011)
• Data: d(x)=x⁵+x⁴+x+1
• So code C(x)=d(x).g(x)
= (x⁵+x⁴+x+1)(x¹⁰+x⁹+x⁸+x⁶+x⁵+x³+1)
= x¹⁵+2x¹⁴+2x¹³+x¹²+2x¹¹+4x¹⁰+3x⁹+2x⁸
+2x⁷+2x⁶+2x⁵+2x⁴+x³+x+1
= x¹⁵+x¹²+ x⁹+ x³+x+1
• Codeword, C: (1001001000001011)

BLOCK DIAGRAM
Block diagram of (15,7) BCH Encoder
Mistake…..????

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