Extraction of TimeSpace Information

1
Chapter 4 Cyclic Codes and Arithmetic Codes
Reading Wakely 80 pp.34-49 Prad 86 pp.299-318
2
Application of Systematic Cyclic Code

• Applied to Magnetic Tapes in CRC (Cyclic
  Redundancy Check) code
• Note there are 9 horizontal tracks, 8 data, one
  parity
• Parity track syndrome used to correct data in
  cyclic code

Byte B0 is cyclic check on B1-B7 in (64, 56)
cyclic code
8 tracks of data arranged vertically as bytes
B0
B1
B2
B3
B4
B5
B7
B6
9th track Vertical parity Check
3
Outline

• Codes for storage and communication
• Cyclic Codes
• Checksum codes
• Codes for arithmetic
• AN Codes
• Residue codes
• Codes for control units (unidirectional errors)
• m-out-of-n codes
• Berger Codes
• Alternate Readings
• Pages 98-125, Johnson89, Pages 94-104, SiSw82
• Pages 299-318, Pradhan

4
Cyclic Codes

• Cyclic codes are parity check codes with
  additional property that cyclic shift of codeword
  also a codeword
• If (Cn-1, Cn-2,, C1, C0) is a codeword, (Cn-2,
  Cn-3,, C0, Cn-1) is also codeword
• Parity check codes require complex encoding,
  decoding circuit using arrays of EX-OR gates, AND
  gates, etc.
• Cyclic codes require far less hardware, in form
  of linear feedback shift registers
• Cyclic codes are used in sequential storage
  devices, e.g. tapes, disks, and data links
• An (n, k) cyclic code can detect single bit
  errors, and multiple adjacent bit error affecting
  fewer than (n-k) bits, burst transient errors.

5
Cyclic Code and Polynomials

• Cyclic codes depend on the representation of data
  by polynomial
• If (Cn-1, Cn-2,, C1, C0) is a codeword, its
  polynomial representation
• C(x) Cn-1xn-1 Cn-2xn-2 C1x1 C0
• Cyclic codes are characterized by its generator
  polynomial g(x)
• g(x) is a polynomial of degree (n-k) for an (n,
  k) code, with unity coefficient in (n-k) term
• Example for (7, 4) code,
• C(x) C6x6 C5x5 C1x1 C0
• g(x) x3 x 1

6
Cyclic And Polynomial Algebra

• Definition A cyclic code is a parity check code,
  where every cyclic shift of a code word is a code
  word, e.g. if (Cn-1, Cn-2,, C1, C0) is a code
  word, then (Cn-2, Cn-3,, C0, Cn-1) is also a
  code word
• Cyclic codes are conveniently described as
  polynomials C(x) Cn-1xn-1 Cn-2xn-2 C1x1
  C0
• Use key concept of polynomial division using
  Euclidean division algorithm f(x) q(x)p(x)
  r(x), degree (r(x)) lt degree (q(x)), q
  quotient and r remainder

7
Cyclic And Polynomial Algebra(Cont.)

• Most important modulo base polynomial for cyclic
  codes is Xn -1 is since 1 Xnmod(Xn-1)
• Now we have

8
Basic Operations on Polynomials

• Can multiply or divide one polynomial by another,
  follow modulo 2 arithmetic, coefficients are 1 or
  0, and addition and subtraction are same.
• Multiplication (x4 x3 x2 1)(x3 x)

Division
x Quotient
x4 x3 x2 1
x5 x4
x5 x4 x3 x
x3 x
Remainder
9
Properties of Generator Polynomials

• g(x) is unique lowest degree nonzero polynomial
  with degree n-k
• Every code polynomial is some multiple of g(x)
• g(x) is a factor of xn - 1, i.e., it divides it
  with zero remainder
• If polynomial with degree n-k divides xn -1, then
  g(x) generated cyclic code

10
Unidirectional Error Model

• Given a bit string (X1X2X3Xn), Xi 0, 1
• Under error, either some 0 ? 1 or 1 ? 0, but not
  both
• at the same time, e.g., if you know that
  s-a-0 faults
• are the only faults that can occur

11
Berger Codes

• Used in Control Units
• Systematic Code
• For an n-component vector, add a check vector
  equal to count of zeros in vector
• If information is X, c(X) number of zeros in X
• Example X lt10010001gt, c(X) 5 101
  so, Code lt10010001101gt
• Detects unidirectional multiple errors

12
Berger Code Error Detection

• If unidirectional errors in information part,
• 1 ? 0 number of zeros in information gt check
  value
• 0 ? 1 number of zeros in information lt check
  value
• If unidirectional error in check part
• 0 ? 1 number of zeros in information lt check
  value
• 1 ? 0 number of zeros in information gt check
  value
• Unidirectional errors in both information and
  check
• parts,
• 1? 0 number of zeros in information increases,
  check value decreases
• 0 ?1 number of zeros in information decreases,
  check value increases

13
M-Out-of-N Codes

• Non-systematic codes
• Used in control units
• An n-bit code word contains exactly
• m-bit 1s. e.g., 3-of-5 code
• Detects all unidirectional
• multiple errors

14
Arithmetic Codes

• Useful to check arithmetic operations
• Parity codes are not preserved under addition,
  subtraction
• Arithmetic codes can be separate (check symbols
  disjoint from data symbols) or nonseparate
  (combined check and data)
• Several Arithmetic codes
• AN codes, Residue codes, Bi-residue codes
• Arithmetic codes have been used in STAR fault
  tolerant computer for space applications

15
Arithmetic Codes

• Useful for integer arithmetic
• Parity-check codes are useful for checking data
  transmission and storage, but are NOT preserved
  by arithmetic operations
• i.e., if A, B ?? S, A ? B ? S

16
Arithmetic Error Model

• Given a binary vector (bn-1,,b0) representing
  an integer N where
• Instead of viewing an error as changing a certain
  number of bits, we consider its effect on the
  integer
• N ? N E
• Absolute value of error is error magnitude.

17
Distance, Weight of Arithmetic Codes

• Definition Arithmetic Weight of an integer I is
  the minimum possible number of nonzero terms in
  an expression of the form
• e.g., 29 (11101)2 25 22 20. so
  w(29) 3
• Definition The weight of an arithmetic error
  which changes N to N E is the arithmetic weight
  of E
  e.g., if each single fault in an adder produces
  an error value of ? 2i for some I, then at least
  k faults are needed to produce an error with
  weight k
• Definition The arithmetic distance between two
  integers N1 and N2 is the arithmetic weight of N2
  - N1

18
Types of Arithmetic Codes

• Separate codes (where information and the check
  are processed separately)
• Encoding of datum X to form f(X) ltc(X), Xgt
  (code word), i.e., concatenation of X and check
  symbol on X
• Furthermore, there are different non-interacting
  arithmetic operations

X
C
f(X) ltC(X), Xgt
Y
C(X)
C
C(Y)
f(Y) ltC(Y), Ygt
19
Types of Arithmetic Codes(Cont.)

• Advantage
• It can perform arithmetic and checking
  operations in parallel
• Nonseparate Codes
• Encoding of datum X ? X f(X)
• Encoding of sum of X and Y is obtained by a
  single binary operation

X
f(X)X
f
Y
f(Y)Y
f
20
AN Codes

• Data X is multiplied by check base A to form
• AX (nonseparate code)
• Addition of code words performed modulo M where A
  divides M
• A
• Check operation by dividing result by A
• If result 0, no error, else error

AX
AY
Residue Mod A
21
AN Code

• There is a cost in number of radix r digits to
  represent a set of number. If NM is the largest
  actual number to be represented, the number of
  digits to be provided must exceed logrANM logrA
  logrNM instead of logrNM.

22
Example of 3N Code
3N CODE B
3N CODE A
Resulting 3N code words for 4-bit information
words Original Information
3N code word 0000
000000 0001 000110
0010 001001 0011
001100 0100
001111 0101
010010 0110 010101
0111 011000 1001
011011 1010
011110 1011
100001 1100 100100
1101 100111 1110
101010 1111
101101
ADDER
3N CODE of Sum
Illustration of the error detection capabilities
of the 3N arithmetic code. The presence of the
fault results in the sum being an invalid 3N code.
23
Residue Codes

• Separate (X, X Mod A) created by appending the
  residue of a number to that number
• Addition of code words performed modulo M, such
  that A divides M

24
Residue Code

• (N1 ? N2) mod A (R1 ? R2) mod A
• (N1N2) mod A (R1R2) mod A
• Resides do not preserve the similar property in
  the division.
• N2 S QN1
• (R2 RS) mod A (RQR1) mod A
• Assume that the sum of two integers N1 and N2
  exceeds the maximum integer M, which the register
  can represent. Then N1 N2 M, instead of N1
  N2 itself, will be placed in the register and the
  residue will be calculate from this content.

25
Residue Code

• The residue thus calculated is not necessary
  equal to (R1 R2) mod A.
• Theorem
• If and only if M is a multiple of A,
• (N1 N2 M) mod A (R1 R2) mod A

26
Low Cost Residue/AN Codes

• Function c(X)X mod A has to be computed in both
  codes
• General division is costly
• Choose A 2a -1
• X mod A ? Xi mod (2a 1), compute through
• tree of adders, Xi is a byte of a-bit width.
  X (Xn-1 Xn-2 X0)

a
a
a
a
a
a
a
a







27
Low Cost Codes

• The function, c(X) X modulo A, must be computed
  in both residue and AN codes
• In general, requires long division ? slow and
  complex
• Choose A 2a -1,then w wi ? (2a)i modulo 2a
  1 for i ? (non-negative integer)

28
Low Cost Codes (Cont.)

• Hence X modulo 2a -1 can be accomplished by
  end-around-carry addition of n a-bit bytes
• Example Consider N (110111010011)2, A 23 1
  7

29
Low Cost Codes (Cont.)
Decimal Binary N
N 5N code Residue code 0
000 000000
(000,000) 1 001
000101 (001,001) 2
010 001010 (010,010)
3 011 001111
(011,011) 4 100
010100 (100,100) 5
101 011001 (101,000)
6 110 011110
(110,001) 7 111
100011 (111,010)
30
Inverse Residue Codes

• Separate code
• Check symbol function
• c(X)A - (X modulo A)
• More effective than residue codes for detecting
  unidirectional multiple errors.
• Operate on X and Y to produce Z (main result)
• Operate on X and Y to produce Z (check
  symbols)
• Checking algorithm computes the expression F
• FZ modulo A Z modulo A
• If F0, no error, else detectable error

31
Summary

• Codes for storage and communication
• Cyclic Codes
• Checksum Codes
• Codes for arithmetic
• AN Codes
• Residue Codes
• Codes for control units (unidirectional errors)
• m-out-of-n Codes
• Berger Codes

32
Time Redundancy

• Time redundancy uses additional time to perform
  the functions of a system such that fault
  detection and often fault tolerance can be
  achieved.
• Attempt to reduce the amount of extra hardware at
  the expense of using additional time.
• In many applications, the time is of much less
  importance than the hardware.

33
Transient Fault Detection

• The basic concept of time redundancy is the
  repetition of computations in ways that allow
  faults to be detected.
• The approach shown in Fig. 3. 62 is often good
  for detecting errors resulting from transient
  faults, but they cannot protect against errors
  resulting from permanent faults.

34
Permanent Fault Detection

• Alternating logic, Recomputing with shifted
  operands (RESO), recomputing with swapped
  operands (RESWO), and recomputing with
  duplication with comparison (REDWC)

35
Alternating Logic

• The alternating logic concept has been applied to
  the transmission of digital data over wire media
  and the detection of faults in digital circuits.
• A combinational circuit is said to be self dual
  if and only if f(X) f(X), where f is the
  output of the circuit and X is the input vector
  for the circuit.
• The full-adder is a self-dual circuit.

36
Alternating Logic

• A key advantage of the alternating logic approach
  is that any combinational circuit with n input
  variables can be transformed into a self-dual
  circuit with no more than n 1 input variables.
• The dual of a function fd is given by
• fd f(x1, x2, , xn)
• The function fsd given by
• fsd xn1f xn1fd
• is then a self-dual function

37
Recomputing with Shifted Operands (RESO)

• RESO was developed as a method to provide
  concurrent error detection in arithmetic logic
  units.
• RESO uses the basic time redundancy method that
  was shown in Fig. 3. 63, and the encoding
  function is selected as the left shift operation
  with the decoding function being the right shift
  operation.

38
Recomputing with Shifted Operands (RESO)
B7
A7
B6
A6
B5
A5
B4
A4
B3
A3
B2
A2
B1
A1
B0
A0
Unshifted Operation at time t0
R7
R6
R5
R4
R3
R2
R1
R0
Faulty Bit Slice
B7
A7
B6
A6
B5
A5
B4
A4
B3
A3
B2
A2
B1
A1
B0
A0
0 0
Shifted Operation at time t0?
R7
R6
R5
R4
R3
R2
R1
R0