7.1 Definitions and Generator Polynomials

Size: px
Start display at page:

Download "7.1 Definitions and Generator Polynomials"

Transcription

1 Chapter 7 Cyclic Codes Lecture 21, March 29, Definitions and Generator Polynomials Cyclic codes are an important class of linear codes for which the encoding and decoding can be efficiently implemented using shift registers. Many common linear codes, including Hamming and Golay codes, have an equivalent cyclic representation. Definition 7.1 (Cyclic Codes). A linear code C is cyclic if (a 0, a 1,..., a n 1 ) C (a n 1, a 0, a 1,..., a n 2 ) C. The word (a n r,..., a n 1, a 0,..., a n r 1 ) is said to be obtained from (a 0, a 1,..., a n 1 ) F n q by cyclically shifting r positions. Remark. to C. 1). If u is a codeword of a cyclic code C, then all cyclic shifts of u also belong 2). We will see that the dual code of a cyclic code is also a cyclic code. Example ). The binary linear code (000, 101, 011, 110) is cyclic. 2). The (7, 16, 3) perfect code in Chapter 1, which we now know is equivalent to Ham(3, 2), is cyclic. 3). The binary linear code (0000, 1001, 0110, 1111) is not cyclic. However, upon interchanging the third and fourth positions, we note that it is equivalent to the linear code (0000, 1010, 0101, 1111), which is cyclic. 69

2 70 Math 422. Coding Theory In order to convert the combinatorial structure of cyclic codes into an algebraic one, we consider the following correspondence: π : Fq n F q x/(x n 1), with (a 0, a 1,..., a n 1 ) a 0 + a 1 x + + a n 1 x n 1. Then π is an F q -linear transformation of vector spaces over F q. From now on, we will sometimes identify Fq n with F q x/(x n 1), and a vector u = (a 0, a 1,..., a n 1 ) with the polynomial u(x) = n 1 a i x i. From Theorem 6.1, we know that F q x/(x n 1) is a ring (but i=0 not a field unless n = 1). Thus, we have a multiplicative operation besides the addition in Fq n. We denote the ring F q x/(x n 1) by Rq n. Example 7.2. Consider the cyclic code C = {000, 110, 101, 011}; then π(c) = {0, 1 + x, 1 + x 2, x + x 2 } F 2 x/(x 3 1) = R2. 3 Remark. A cyclic code in Fq n can be thought of as a particular subset of the residue class polynomial ring Rq n. A codeword u = (a 0, a 1,..., a n 1 ) in a cyclic code C with the polynomial u(x) = a 0 + a 1 x + + a n 1 x n 1. Then (a n 1, a 0, a 1,..., a n 2 ) corresponds to the polynomial v(x) = a n 1 + a 0 x + a 1 x a n 2 x n 1 = xu(x)(mod x n 1), Thus, a linear code C is cyclic iff u(x) C = xu(x)(mod x n 1) C. That is, multiplication by x (modulo the polynomial x n 1) corresponds to a cyclic shift. Definition (Ideals). Let R be a ring. A nonempty subset I of R is called an ideal if 1). Both a + b and a b belong to I, for all a, b I; 2). r a I, for all r R and a I. Examples. In the ring F 2 x/(x 3 1) = R2, 3 then subset I = {0, 1 + x, 1 + x 2, x + x 2 } is an ideal. In fact, the following theorem shows that a cyclic code C is an ideal of R n q. Theorem 7.1 (Cyclic Codes are Ideals). A linear code C in R n q is cyclic. u(x) C, r(x) R n q = r(x)u(x) C.

3 7.2. Generator and parity-check matrices 71 Definition 7.2 (Principal ideal). An ideal I of a ring R is called a principal ideal if there exists an element g I such that I = g = {r g r R}. The element g is called a generator of I and I is said to be generated by g. A ring R is called a principal ideal ring if every ideal of R is principal. The next theorem states that every ideal in Rq n is a principal ideal (i.e. Rq n is a Principal Ideal Ring). Theorem 7.2 (Generator Polynomial). Let C be a nonzero q-ary cyclic code in Rq n. Then 1). there exists a unique monic polynomial g(x) of smallest degree in C; 2). C = g(x) ; 3). g(x) is a factor of x n 1 in F q x. Definition 7.3 (Generator polynomial). The monic polynomial of least degree in the above Theorem is called the generator polynomial of the cyclic code C. Theorem 7.3 (Lowest Generator Polynomial Coefficient). Let g(x) = g 0 + g 1 x + + g r 1 x r 1 + x r be the generator polynomial of a cyclic code C. Then g 0 0. Proof. Suppose g 0 = 0. Then x n 1 g(x) = x 1 g(x) R n q is a codeword of degree r 1, contradicting the minimality of deg(g(x)). Lecture 22, March 31, Generator and parity-check matrices In the previous section, we showed that a cyclic code is totally determined by its generator polynomial. Hence, such a code should also have generator matrices determined by this polynomial. Indeed, we have the following result.

4 72 Math 422. Coding Theory Theorem 7.4 (Cyclic Generator Matrix). A cyclic code C with generator polynomial g(x) = g 0 + g 1 x + + g r x r has dimension n r and generator matrix g(x) g 0 g 1 g r xg(x) 0 g 0 g 1 g r 0 0 G =. = x n r 2 g(x) 0 0 g 0 g 1 g r 0 x n r 1 g(x) g 0 g 1 g r Note that we identify a vector with a polynomial. Proof. Let u(x) be a codeword in a cyclic code C with generator polynomial g(x). From Theorem 7.2, we know that C = g(x), i.e., u(x) = q(x)g(x) for some polynomial q(x). Note that deg(q(x)) < n r since deg(u(x)) < n. That is, u(x) = (q 0 + q 1 x + + q n r 1 x n r 1 )g(x) = q 0 g(x) + q 1 xg(x) + + q n r 1 x n r 1 g(x), which is a linear combination of the n r rows g(x), xg(x),..., x n r 1 g(x) of G. The nonzero g 0 s ensure that the rows of G are linearly independent. Thus, the span of the rows of G is the n r dimensional code C. Remark. Together, the above Theorems say that an n, k code is cyclic. it is generated by a factor of x n 1 with degree n k, i.e., there is a one-to-one correspondence between the cyclic codes in Fq n and the monic divisors of x n 1 F q x. Example 7.3 (All ternary cyclic codes of length 4). Suppose we wish to find all ternary cyclic codes of length n = 4. The generators for such codes must be factors of x 4 1 in the ring F 3 x. Since 1 is a root of the equation x 4 1 we know that (x 1) is a factor and hence (x 4 1) = (x 1)(x 3 + x 2 + x + 1) It is clear that the factor x 3 + x 2 + x + 1 is not irreducible over F 3 because it has a root 2 in F 3. Using long division, we obtain (x 4 1) = (x 1)(x + 1)(x 2 + 1). Since any combination of these three irreducible factors can be used to construct a generator polynomial g(x) for a cyclic code, there are a total of 2 3 = 8 ternary cyclic codes of length 4, as illustrated in the following Table. Upon examining the weights of the rows of the possible generator matrices, we see that the generated codes either have minimum distance less than or equal to 2 or else equal to

5 7.2. Generator and parity-check matrices Hence, it is not possible to have a cyclic code of length 4 and minimum distance 3. In particular, Ham(2, 3), for which n = (3 2 1)/(3 1) = 4, cannot be cyclic. Thus, not all Hamming codes have a cyclic representation. Table 7.1: GP g(x) and corresponding GM G for all ternary cyclic codes of length 4. g(x) 1 x 1 x + 1 x (x 1)(x + 1) = x 2 1 (x 1)(x 2 + 1) = x 3 x 2 + x 1 (x + 1)(x 2 + 1) = x 3 + x 2 + x + 1 x 4 1 = 0 G Definition 7.4 (Check Polynomial). Let C be a cyclic n, k code with generator polynomial g(x). Then the polynomial h(x) = (x n 1)/g(x) is called the check polynomial of C. Since g(x) is monic and degree n k, we see that h(x) is monic and degree k. Theorem 7.5 (Cyclic Check Polynomial). An element u(x) of Rq n is a codeword of the cyclic code with check polynomial h(x). u(x)h(x) = 0 Rq n. Proof. If u(x) is a codeword, then in R n q we have u(x) = a(x)g(x) u(x)h(x) = a(x)g(x)h(x) = a(x)(x n 1) = 0 R n q. We can express any polynomial u(x) in Rq n as u(x) = q(x)g(x) + r(x) where deg(r(x)) < deg(g(x)) = n k. If u(x)h(x) = 0 then r(x)h(x) = u(x)h(x) q(x)g(x)h(x) =

6 74 Math 422. Coding Theory 0 Rq n. But deg(r(x)h(x)) < n k + k = n, so r(x)h(x) = 0 in F q x, not just in Rq n. If r(x) 0, consider its highest degree coefficient a 0. Then since h(x) is monic, the coefficient of the highest degree term of the product r(x)h(x) is a 0, which is a contradiction. Thus r(x) = 0 and so u(x) = q(x)g(x) g(x) is a codeword. Theorem 7.6 (Cyclic Parity-Check Matrix). A cyclic code with check polynomial h(x) = h 0 + h 1 x + + h k x k has dimension k and a parity check matrix h k h k 1 h h k h k 1 h H = h k h k 1 h h k h k 1 h 0 Proof. Since the degree of the generator polynomial g(x) is r = n k, by Theorem 7.4, the dimension of the code must be k. From Theorem 7.5, we know that a codeword u(x) = c 0 + c 1 x + + c n 1 x n 1 must satisfy u(x)h(x) = 0. In particular, the coefficients of x k, x k+1,..., x n 1 of the product u(x)h(x) must be zero; for l = k, k + 1,..., n 1 we then have 0 = c i h j. i+j=l But then, since each of these equations is one of the n k rows of the matrix equation h k h k 1 h c h k h k 1 h c =. 0 0 h k h k 1 h h k h k 1 h 0 the codewords are orthogonal to all cyclic shifts of the vector h k h k 1 h k 2 h The codewords are thus orthogonal to all linear combinations of the rows of H. This means that C contains the span of the rows of H. But h k = 1, so we see that H has rank n k and hence generates exactly the linear subspace C. That is, H is a parity check matrix for the code with check polynomial h(x). Definition 7.5 (Reciprocal polynomial). Let h(x) = h 0 + h 1 x + + h k x k be a polynomial of degree k (h k 0) over F q. Define the reciprocal polynomial h R (x) of h(x) by h R (x) = x k h(x 1 ) = h 0 x k + h 1 x k h k = h k + h k 1 x + + h 0 x k. c n 2 c n 1 0

7 7.2. Generator and parity-check matrices 75 Remark. For the GP g(x) and CP h(x), since h R (x)g R (x) = x k h(x 1 )x n k g(x 1 ) = x n h(x 1 )g(x 1 ) = x n ((x 1 ) n 1) = 1 x n, we see that h(x) is also a factor of x n 1. In view of above Theorems, this says that C is a cyclic code, with (monic) generator h 1 0 h R (x). Lecture 23, April 5, 2011 We can show that all binary Hamming codes have an equivalent cyclic representation. Theorem 7.7 (Cyclic Binary Hamming Codes). The binary Hamming code Ham(r, 2) is equivalent to a cyclic code. Proof. Let α be a primitive element of F 2 r. Then by Theorem 6.6 the minimal polynomial of α is f(x) = M (1) (x) = j C 1 (x α j ), where C 1 is the cyclotomic coset of 2 modulo 2 r 1 containing 1. Indeed C 1 = {1, 2, 2 2, 2 3,..., 2 r 2, 2 r 1 }, hence deg(f(x)) = r and it is irreducible. By Theorem we know {0, 1, α,..., α 2r 2 } = F 2 r = F 2 x/f(x) = F 2 α = {a 0 +a 1 α+a 2 α 2 + +a r 1 α r 1 a i F 2 }. We associate each element a 0 + a 1 α + a 2 α a r 1 α r 1 F 2 r = {0, 1, α,..., α 2r 2 } with the column vector Let n = 2 r 1. The r n matrix a 0 a 1. a r 1 F 2 r. H = 1, α, α 2,..., α n 1 is seen to be the parity check matrix for C = Ham(r, 2) since its columns are precisely the distinct nonzero vectors of F r 2. A codeword u(x) = c 0 + c 1 x + + c n 1 x n 1 in this code must then satisfy the vector equation c 0 + c 1 α 1 + c 2 α c n 1 α n 1 = 0, so that C = {u(x) R n 2 u(α) = 0 F 2 x/f(x)} = {x F n 2 Hx t = 0}. If u(x) C and r(x) R n 2, we have r(α)u(α) = r(α)0 = 0 F 2 x/f(x), noting that α n = 1, so r(x)u(x) is also an element of C. Theorem 6.1 then implies that C is cyclic.

8 76 Math 422. Coding Theory Example 7.4. The irreducible polynomial x 3 + x + 1 in F 2 x can be used to generate the field F 8 = F 2 x/(x 3 + x + 1) with 2 3 = 8 elements. Note that F 8 has α as a primitive element if α is a root of x 3 + x + 1. Indeed all elements in F 2 α can be expressed as powers of α: F 8 = {0, 1, α, α 2, α 3 = α + 1, α 4 = α 2 + α, α 5 = α 2 + α + 1, α 6 = α 2 + 1}. Note that α 7 = α 3 + α = 1; that is, the primitive element α has order 7 = 8 1. A parity check matrix for a cyclic version of the Hamming code Ham(3, 2) is thus H = Definition 7.6 (Primitive polynomial). A primitive polynomial of F p r irreducible polynomial in F p x having a primitive element of F p r as a root. is a monic Corollary 7.7 (Binary Hamming Generator Polynomials). Any primitive polynomial of F 2 r is a generator polynomial for a cyclic Hamming code Ham(r, 2). Proof. Let α be a primitive element of F 2 r. Its minimal polynomial p(x) is a primitive polynomial of F 2 r. From the proof of Theorem 7.7, we see that Ham(r, 2) consists precisely of those polynomials u(x) for which u(α) = 0, i.e., Ham(r, 2) = {u(x) R 2r 1 2 u(α) = 0} for example, f(x) itself. By Theorem 6.4, any such polynomial must be a multiple of f(x). That is, Ham(r, 2) f(x). Moreover, Theorem 7.1 implies that every multiple of the codeword f(x) belongs to the cyclic code Ham(r, 2). Hence Ham(r, 2) = f(x). Example 7.5. Consider the irreducible polynomial p(x) = x 3 + x + 1 in F 2 x and let α be a root of p(x). Since α is a primitive element of F 8 = F 2 α = F 2 x/p(x) and p(α) = 0, we know that p(x) is a primitive polynomial of F 2 3 = F 2 x/p(x) and hence Ham(3, 2) = p(x). From Theorem 7.4, we can then immediately write down a generator matrix for a cyclic Ham(3, 2) code: G =

3. Coding theory 3.1. Basic concepts

3. Coding theory 3.1. Basic concepts 3. CODING THEORY 1 3. Coding theory 3.1. Basic concepts In this chapter we will discuss briefly some aspects of error correcting codes. The main problem is that if information is sent via a noisy channel,

More information

Outline. MSRI-UP 2009 Coding Theory Seminar, Week 2. The definition. Link to polynomials

Outline. MSRI-UP 2009 Coding Theory Seminar, Week 2. The definition. Link to polynomials Outline MSRI-UP 2009 Coding Theory Seminar, Week 2 John B. Little Department of Mathematics and Computer Science College of the Holy Cross Cyclic Codes Polynomial Algebra More on cyclic codes Finite fields

More information

Generator Matrix. Theorem 6: If the generator polynomial g(x) of C has degree n-k then C is an [n,k]-cyclic code. If g(x) = a 0. a 1 a n k 1.

Generator Matrix. Theorem 6: If the generator polynomial g(x) of C has degree n-k then C is an [n,k]-cyclic code. If g(x) = a 0. a 1 a n k 1. Cyclic Codes II Generator Matrix We would now like to consider how the ideas we have previously discussed for linear codes are interpreted in this polynomial version of cyclic codes. Theorem 6: If the

More information

Linear Cyclic Codes. Polynomial Word 1 + x + x x 4 + x 5 + x x + x f(x) = q(x)h(x) + r(x),

Linear Cyclic Codes. Polynomial Word 1 + x + x x 4 + x 5 + x x + x f(x) = q(x)h(x) + r(x), Coding Theory Massoud Malek Linear Cyclic Codes Polynomial and Words A polynomial of degree n over IK is a polynomial p(x) = a 0 + a 1 + + a n 1 x n 1 + a n x n, where the coefficients a 1, a 2,, a n are

More information

Information Theory. Lecture 7

Information Theory. Lecture 7 Information Theory Lecture 7 Finite fields continued: R3 and R7 the field GF(p m ),... Cyclic Codes Intro. to cyclic codes: R8.1 3 Mikael Skoglund, Information Theory 1/17 The Field GF(p m ) π(x) irreducible

More information

Linear Cyclic Codes. Polynomial Word 1 + x + x x 4 + x 5 + x x + x

Linear Cyclic Codes. Polynomial Word 1 + x + x x 4 + x 5 + x x + x Coding Theory Massoud Malek Linear Cyclic Codes Polynomial and Words A polynomial of degree n over IK is a polynomial p(x) = a 0 + a 1 x + + a n 1 x n 1 + a n x n, where the coefficients a 0, a 1, a 2,,

More information

B. Cyclic Codes. Primitive polynomials are the generator polynomials of cyclic codes.

B. Cyclic Codes. Primitive polynomials are the generator polynomials of cyclic codes. B. Cyclic Codes A cyclic code is a linear block code with the further property that a shift of a codeword results in another codeword. These are based on polynomials whose elements are coefficients from

More information

ECEN 604: Channel Coding for Communications

ECEN 604: Channel Coding for Communications ECEN 604: Channel Coding for Communications Lecture: Introduction to Cyclic Codes Henry D. Pfister Department of Electrical and Computer Engineering Texas A&M University ECEN 604: Channel Coding for Communications

More information

x n k m(x) ) Codewords can be characterized by (and errors detected by): c(x) mod g(x) = 0 c(x)h(x) = 0 mod (x n 1)

x n k m(x) ) Codewords can be characterized by (and errors detected by): c(x) mod g(x) = 0 c(x)h(x) = 0 mod (x n 1) Cyclic codes: review EE 387, Notes 15, Handout #26 A cyclic code is a LBC such that every cyclic shift of a codeword is a codeword. A cyclic code has generator polynomial g(x) that is a divisor of every

More information

Know the meaning of the basic concepts: ring, field, characteristic of a ring, the ring of polynomials R[x].

Know the meaning of the basic concepts: ring, field, characteristic of a ring, the ring of polynomials R[x]. The second exam will be on Friday, October 28, 2. It will cover Sections.7,.8, 3., 3.2, 3.4 (except 3.4.), 4. and 4.2 plus the handout on calculation of high powers of an integer modulo n via successive

More information

: Coding Theory. Notes by Assoc. Prof. Dr. Patanee Udomkavanich October 30, upattane

: Coding Theory. Notes by Assoc. Prof. Dr. Patanee Udomkavanich October 30, upattane 2301532 : Coding Theory Notes by Assoc. Prof. Dr. Patanee Udomkavanich October 30, 2006 http://pioneer.chula.ac.th/ upattane Chapter 1 Error detection, correction and decoding 1.1 Basic definitions and

More information

Chapter 5. Cyclic Codes

Chapter 5. Cyclic Codes Wireless Information Transmission System Lab. Chapter 5 Cyclic Codes Institute of Communications Engineering National Sun Yat-sen University Outlines Description of Cyclic Codes Generator and Parity-Check

More information

EE 229B ERROR CONTROL CODING Spring 2005

EE 229B ERROR CONTROL CODING Spring 2005 EE 9B ERROR CONTROL CODING Spring 005 Solutions for Homework 1. (Weights of codewords in a cyclic code) Let g(x) be the generator polynomial of a binary cyclic code of length n. (a) Show that if g(x) has

More information

A family Of Cyclic Codes Over Finite Chain Rings

A family Of Cyclic Codes Over Finite Chain Rings The Islamic University of Gaza Deanery of Higher Studies Faculty of Science Department of Mathematics A family Of Cyclic Codes Over Finite Chain Rings Presented by: Sanaa Yusuf Sabouh Supervised by: Dr.:

More information

ELG 5372 Error Control Coding. Lecture 12: Ideals in Rings and Algebraic Description of Cyclic Codes

ELG 5372 Error Control Coding. Lecture 12: Ideals in Rings and Algebraic Description of Cyclic Codes ELG 5372 Error Control Coding Lecture 12: Ideals in Rings and Algebraic Description of Cyclic Codes Quotient Ring Example + Quotient Ring Example Quotient Ring Recall the quotient ring R={,,, }, where

More information

MATH3302 Coding Theory Problem Set The following ISBN was received with a smudge. What is the missing digit? x9139 9

MATH3302 Coding Theory Problem Set The following ISBN was received with a smudge. What is the missing digit? x9139 9 Problem Set 1 These questions are based on the material in Section 1: Introduction to coding theory. You do not need to submit your answers to any of these questions. 1. The following ISBN was received

More information

Coding Theory and Applications. Solved Exercises and Problems of Cyclic Codes. Enes Pasalic University of Primorska Koper, 2013

Coding Theory and Applications. Solved Exercises and Problems of Cyclic Codes. Enes Pasalic University of Primorska Koper, 2013 Coding Theory and Applications Solved Exercises and Problems of Cyclic Codes Enes Pasalic University of Primorska Koper, 2013 Contents 1 Preface 3 2 Problems 4 2 1 Preface This is a collection of solved

More information

Rings. EE 387, Notes 7, Handout #10

Rings. EE 387, Notes 7, Handout #10 Rings EE 387, Notes 7, Handout #10 Definition: A ring is a set R with binary operations, + and, that satisfy the following axioms: 1. (R, +) is a commutative group (five axioms) 2. Associative law for

More information

Fault Tolerance & Reliability CDA Chapter 2 Cyclic Polynomial Codes

Fault Tolerance & Reliability CDA Chapter 2 Cyclic Polynomial Codes Fault Tolerance & Reliability CDA 5140 Chapter 2 Cyclic Polynomial Codes - cylic code: special type of parity check code such that every cyclic shift of codeword is a codeword - for example, if (c n-1,

More information

MATH32031: Coding Theory Part 15: Summary

MATH32031: Coding Theory Part 15: Summary MATH32031: Coding Theory Part 15: Summary 1 The initial problem The main goal of coding theory is to develop techniques which permit the detection of errors in the transmission of information and, if necessary,

More information

ERROR CORRECTING CODES

ERROR CORRECTING CODES ERROR CORRECTING CODES To send a message of 0 s and 1 s from my computer on Earth to Mr. Spock s computer on the planet Vulcan we use codes which include redundancy to correct errors. n q Definition. A

More information

Cyclic codes: overview

Cyclic codes: overview Cyclic codes: overview EE 387, Notes 14, Handout #22 A linear block code is cyclic if the cyclic shift of a codeword is a codeword. Cyclic codes have many advantages. Elegant algebraic descriptions: c(x)

More information

EE512: Error Control Coding

EE512: Error Control Coding EE512: Error Control Coding Solution for Assignment on Cyclic Codes March 22, 2007 1. A cyclic code, C, is an ideal genarated by its minimal degree polynomial, g(x). C = < g(x) >, = {m(x)g(x) : m(x) is

More information

+ 1 3 x2 2x x3 + 3x 2 + 0x x x2 2x + 3 4

+ 1 3 x2 2x x3 + 3x 2 + 0x x x2 2x + 3 4 Math 4030-001/Foundations of Algebra/Fall 2017 Polynomials at the Foundations: Rational Coefficients The rational numbers are our first field, meaning that all the laws of arithmetic hold, every number

More information

Finite Fields. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay

Finite Fields. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay 1 / 25 Finite Fields Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay September 25, 2014 2 / 25 Fields Definition A set F together

More information

Algebra Review 2. 1 Fields. A field is an extension of the concept of a group.

Algebra Review 2. 1 Fields. A field is an extension of the concept of a group. Algebra Review 2 1 Fields A field is an extension of the concept of a group. Definition 1. A field (F, +,, 0 F, 1 F ) is a set F together with two binary operations (+, ) on F such that the following conditions

More information

CYCLIC SIEVING FOR CYCLIC CODES

CYCLIC SIEVING FOR CYCLIC CODES CYCLIC SIEVING FOR CYCLIC CODES ALEX MASON, VICTOR REINER, SHRUTHI SRIDHAR Abstract. These are notes on a preliminary follow-up to a question of Jim Propp, about cyclic sieving of cyclic codes. We show

More information

EE512: Error Control Coding

EE512: Error Control Coding EE51: Error Control Coding Solution for Assignment on BCH and RS Codes March, 007 1. To determine the dimension and generator polynomial of all narrow sense binary BCH codes of length n = 31, we have to

More information

G Solution (10 points) Using elementary row operations, we transform the original generator matrix as follows.

G Solution (10 points) Using elementary row operations, we transform the original generator matrix as follows. EE 387 October 28, 2015 Algebraic Error-Control Codes Homework #4 Solutions Handout #24 1. LBC over GF(5). Let G be a nonsystematic generator matrix for a linear block code over GF(5). 2 4 2 2 4 4 G =

More information

: Error Correcting Codes. November 2017 Lecture 2

: Error Correcting Codes. November 2017 Lecture 2 03683072: Error Correcting Codes. November 2017 Lecture 2 Polynomial Codes and Cyclic Codes Amnon Ta-Shma and Dean Doron 1 Polynomial Codes Fix a finite field F q. For the purpose of constructing polynomial

More information

Abstract Algebra: Chapters 16 and 17

Abstract Algebra: Chapters 16 and 17 Study polynomials, their factorization, and the construction of fields. Chapter 16 Polynomial Rings Notation Let R be a commutative ring. The ring of polynomials over R in the indeterminate x is the set

More information

Polynomials. Chapter 4

Polynomials. Chapter 4 Chapter 4 Polynomials In this Chapter we shall see that everything we did with integers in the last Chapter we can also do with polynomials. Fix a field F (e.g. F = Q, R, C or Z/(p) for a prime p). Notation

More information

The Golay codes. Mario de Boer and Ruud Pellikaan

The Golay codes. Mario de Boer and Ruud Pellikaan The Golay codes Mario de Boer and Ruud Pellikaan Appeared in Some tapas of computer algebra (A.M. Cohen, H. Cuypers and H. Sterk eds.), Project 7, The Golay codes, pp. 338-347, Springer, Berlin 1999, after

More information

Lecture 7: Polynomial rings

Lecture 7: Polynomial rings Lecture 7: Polynomial rings Rajat Mittal IIT Kanpur You have seen polynomials many a times till now. The purpose of this lecture is to give a formal treatment to constructing polynomials and the rules

More information

Coding Theory: Linear-Error Correcting Codes Anna Dovzhik Math 420: Advanced Linear Algebra Spring 2014

Coding Theory: Linear-Error Correcting Codes Anna Dovzhik Math 420: Advanced Linear Algebra Spring 2014 Anna Dovzhik 1 Coding Theory: Linear-Error Correcting Codes Anna Dovzhik Math 420: Advanced Linear Algebra Spring 2014 Sharing data across channels, such as satellite, television, or compact disc, often

More information

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra EE 387, Notes 13, Handout #20 Fundamental Theorem of Algebra Lemma: If f(x) is a polynomial over GF(q) GF(Q), then β is a zero of f(x) if and only if x β is a divisor of f(x). Proof: By the division algorithm,

More information

: Error Correcting Codes. October 2017 Lecture 1

: Error Correcting Codes. October 2017 Lecture 1 03683072: Error Correcting Codes. October 2017 Lecture 1 First Definitions and Basic Codes Amnon Ta-Shma and Dean Doron 1 Error Correcting Codes Basics Definition 1. An (n, K, d) q code is a subset of

More information

Solutions of Exam Coding Theory (2MMC30), 23 June (1.a) Consider the 4 4 matrices as words in F 16

Solutions of Exam Coding Theory (2MMC30), 23 June (1.a) Consider the 4 4 matrices as words in F 16 Solutions of Exam Coding Theory (2MMC30), 23 June 2016 (1.a) Consider the 4 4 matrices as words in F 16 2, the binary vector space of dimension 16. C is the code of all binary 4 4 matrices such that the

More information

Cyclic codes. Vahid Meghdadi Reference: Error Correction Coding by Todd K. Moon. February 2008

Cyclic codes. Vahid Meghdadi Reference: Error Correction Coding by Todd K. Moon. February 2008 Cyclic codes Vahid Meghdadi Reference: Error Correction Coding by Todd K. Moon February 2008 1 Definitions Definition 1. A ring < R, +,. > is a set R with two binary operation + (addition) and. (multiplication)

More information

ϕ : Z F : ϕ(t) = t 1 =

ϕ : Z F : ϕ(t) = t 1 = 1. Finite Fields The first examples of finite fields are quotient fields of the ring of integers Z: let t > 1 and define Z /t = Z/(tZ) to be the ring of congruence classes of integers modulo t: in practical

More information

Negacyclic and Constacyclic codes over finite chain rings June 29, 2008

Negacyclic and Constacyclic codes over finite chain rings June 29, 2008 Negacyclic and Constacyclic rings codes over finite chain June 29, 2008 THE ISLAMIC UNIVERSITY OF GAZA DEANERY OF HIGHER STUDIES FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Negacyclic and Constacyclic

More information

AN INTRODUCTION TO THE THEORY OF FIELD EXTENSIONS

AN INTRODUCTION TO THE THEORY OF FIELD EXTENSIONS AN INTRODUCTION TO THE THEORY OF FIELD EXTENSIONS SAMUEL MOY Abstract. Assuming some basic knowledge of groups, rings, and fields, the following investigation will introduce the reader to the theory of

More information

Polynomial Rings. (Last Updated: December 8, 2017)

Polynomial Rings. (Last Updated: December 8, 2017) Polynomial Rings (Last Updated: December 8, 2017) These notes are derived primarily from Abstract Algebra, Theory and Applications by Thomas Judson (16ed). Most of this material is drawn from Chapters

More information

The extended Golay code

The extended Golay code The extended Golay code N. E. Straathof July 6, 2014 Master thesis Mathematics Supervisor: Dr R. R. J. Bocklandt Korteweg-de Vries Instituut voor Wiskunde Faculteit der Natuurwetenschappen, Wiskunde en

More information

Lecture 17: Perfect Codes and Gilbert-Varshamov Bound

Lecture 17: Perfect Codes and Gilbert-Varshamov Bound Lecture 17: Perfect Codes and Gilbert-Varshamov Bound Maximality of Hamming code Lemma Let C be a code with distance 3, then: C 2n n + 1 Codes that meet this bound: Perfect codes Hamming code is a perfect

More information

Cyclic codes. I give an example of a shift register with four storage elements and two binary adders.

Cyclic codes. I give an example of a shift register with four storage elements and two binary adders. Good afternoon, gentleman! Today I give you a lecture about cyclic codes. This lecture consists of three parts: I Origin and definition of cyclic codes ;? how to find cyclic codes: The Generator Polynomial

More information

Repeated-Root Self-Dual Negacyclic Codes over Finite Fields

Repeated-Root Self-Dual Negacyclic Codes over Finite Fields Journal of Mathematical Research with Applications May, 2016, Vol. 36, No. 3, pp. 275 284 DOI:10.3770/j.issn:2095-2651.2016.03.004 Http://jmre.dlut.edu.cn Repeated-Root Self-Dual Negacyclic Codes over

More information

The BCH Bound. Background. Parity Check Matrix for BCH Code. Minimum Distance of Cyclic Codes

The BCH Bound. Background. Parity Check Matrix for BCH Code. Minimum Distance of Cyclic Codes S-723410 BCH and Reed-Solomon Codes 1 S-723410 BCH and Reed-Solomon Codes 3 Background The algebraic structure of linear codes and, in particular, cyclic linear codes, enables efficient encoding and decoding

More information

Cyclic Codes. Saravanan Vijayakumaran August 26, Department of Electrical Engineering Indian Institute of Technology Bombay

Cyclic Codes. Saravanan Vijayakumaran August 26, Department of Electrical Engineering Indian Institute of Technology Bombay 1 / 25 Cyclic Codes Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 26, 2014 2 / 25 Cyclic Codes Definition A cyclic shift

More information

Math 512 Syllabus Spring 2017, LIU Post

Math 512 Syllabus Spring 2017, LIU Post Week Class Date Material Math 512 Syllabus Spring 2017, LIU Post 1 1/23 ISBN, error-detecting codes HW: Exercises 1.1, 1.3, 1.5, 1.8, 1.14, 1.15 If x, y satisfy ISBN-10 check, then so does x + y. 2 1/30

More information

5.0 BCH and Reed-Solomon Codes 5.1 Introduction

5.0 BCH and Reed-Solomon Codes 5.1 Introduction 5.0 BCH and Reed-Solomon Codes 5.1 Introduction A. Hocquenghem (1959), Codes correcteur d erreurs; Bose and Ray-Chaudhuri (1960), Error Correcting Binary Group Codes; First general family of algebraic

More information

MATH 291T CODING THEORY

MATH 291T CODING THEORY California State University, Fresno MATH 291T CODING THEORY Spring 2009 Instructor : Stefaan Delcroix Chapter 1 Introduction to Error-Correcting Codes It happens quite often that a message becomes corrupt

More information

Homework 8 Solutions to Selected Problems

Homework 8 Solutions to Selected Problems Homework 8 Solutions to Selected Problems June 7, 01 1 Chapter 17, Problem Let f(x D[x] and suppose f(x is reducible in D[x]. That is, there exist polynomials g(x and h(x in D[x] such that g(x and h(x

More information

Duadic Codes over Finite Commutative Rings

Duadic Codes over Finite Commutative Rings The Islamic University of Gaza Faculty of Science Department of Mathematics Duadic Codes over Finite Commutative Rings PRESENTED BY Ikhlas Ibraheem Diab Al-Awar SUPERVISED BY Prof. Mohammed Mahmoud AL-Ashker

More information

MATH Examination for the Module MATH-3152 (May 2009) Coding Theory. Time allowed: 2 hours. S = q

MATH Examination for the Module MATH-3152 (May 2009) Coding Theory. Time allowed: 2 hours. S = q MATH-315201 This question paper consists of 6 printed pages, each of which is identified by the reference MATH-3152 Only approved basic scientific calculators may be used. c UNIVERSITY OF LEEDS Examination

More information

We saw in the last chapter that the linear Hamming codes are nontrivial perfect codes.

We saw in the last chapter that the linear Hamming codes are nontrivial perfect codes. Chapter 5 Golay Codes Lecture 16, March 10, 2011 We saw in the last chapter that the linear Hamming codes are nontrivial perfect codes. Question. Are there any other nontrivial perfect codes? Answer. Yes,

More information

MTH310 EXAM 2 REVIEW

MTH310 EXAM 2 REVIEW MTH310 EXAM 2 REVIEW SA LI 4.1 Polynomial Arithmetic and the Division Algorithm A. Polynomial Arithmetic *Polynomial Rings If R is a ring, then there exists a ring T containing an element x that is not

More information

Polynomial Rings. i=0

Polynomial Rings. i=0 Polynomial Rings 4-15-2018 If R is a ring, the ring of polynomials in x with coefficients in R is denoted R[x]. It consists of all formal sums a i x i. Here a i = 0 for all but finitely many values of

More information

1 Vandermonde matrices

1 Vandermonde matrices ECE 771 Lecture 6 BCH and RS codes: Designer cyclic codes Objective: We will begin with a result from linear algebra regarding Vandermonde matrices This result is used to prove the BCH distance properties,

More information

CHAPTER 14. Ideals and Factor Rings

CHAPTER 14. Ideals and Factor Rings CHAPTER 14 Ideals and Factor Rings Ideals Definition (Ideal). A subring A of a ring R is called a (two-sided) ideal of R if for every r 2 R and every a 2 A, ra 2 A and ar 2 A. Note. (1) A absorbs elements

More information

Coding Theory ( Mathematical Background I)

Coding Theory ( Mathematical Background I) N.L.Manev, Lectures on Coding Theory (Maths I) p. 1/18 Coding Theory ( Mathematical Background I) Lector: Nikolai L. Manev Institute of Mathematics and Informatics, Sofia, Bulgaria N.L.Manev, Lectures

More information

Chapter 6 Reed-Solomon Codes. 6.1 Finite Field Algebra 6.2 Reed-Solomon Codes 6.3 Syndrome Based Decoding 6.4 Curve-Fitting Based Decoding

Chapter 6 Reed-Solomon Codes. 6.1 Finite Field Algebra 6.2 Reed-Solomon Codes 6.3 Syndrome Based Decoding 6.4 Curve-Fitting Based Decoding Chapter 6 Reed-Solomon Codes 6. Finite Field Algebra 6. Reed-Solomon Codes 6.3 Syndrome Based Decoding 6.4 Curve-Fitting Based Decoding 6. Finite Field Algebra Nonbinary codes: message and codeword symbols

More information

Foundations of Cryptography

Foundations of Cryptography Foundations of Cryptography Ville Junnila viljun@utu.fi Department of Mathematics and Statistics University of Turku 2015 Ville Junnila viljun@utu.fi Lecture 7 1 of 18 Cosets Definition 2.12 Let G be a

More information

Lecture 14: Hamming and Hadamard Codes

Lecture 14: Hamming and Hadamard Codes CSCI-B69: A Theorist s Toolkit, Fall 6 Oct 6 Lecture 4: Hamming and Hadamard Codes Lecturer: Yuan Zhou Scribe: Kaiyuan Zhu Recap Recall from the last lecture that error-correcting codes are in fact injective

More information

Hamming codes and simplex codes ( )

Hamming codes and simplex codes ( ) Chapter 6 Hamming codes and simplex codes (2018-03-17) Synopsis. Hamming codes are essentially the first non-trivial family of codes that we shall meet. We start by proving the Distance Theorem for linear

More information

Irreducible Polynomials. Finite Fields of Order p m (1) Primitive Polynomials. Finite Fields of Order p m (2)

Irreducible Polynomials. Finite Fields of Order p m (1) Primitive Polynomials. Finite Fields of Order p m (2) S-72.3410 Finite Fields (2) 1 S-72.3410 Finite Fields (2) 3 Irreducible Polynomials Finite Fields of Order p m (1) The following results were discussed in the previous lecture: The order of a finite field

More information

Groups, Rings, and Finite Fields. Andreas Klappenecker. September 12, 2002

Groups, Rings, and Finite Fields. Andreas Klappenecker. September 12, 2002 Background on Groups, Rings, and Finite Fields Andreas Klappenecker September 12, 2002 A thorough understanding of the Agrawal, Kayal, and Saxena primality test requires some tools from algebra and elementary

More information

(Rgs) Rings Math 683L (Summer 2003)

(Rgs) Rings Math 683L (Summer 2003) (Rgs) Rings Math 683L (Summer 2003) We will first summarise the general results that we will need from the theory of rings. A unital ring, R, is a set equipped with two binary operations + and such that

More information

MATH 291T CODING THEORY

MATH 291T CODING THEORY California State University, Fresno MATH 291T CODING THEORY Fall 2011 Instructor : Stefaan Delcroix Contents 1 Introduction to Error-Correcting Codes 3 2 Basic Concepts and Properties 6 2.1 Definitions....................................

More information

ECEN 5682 Theory and Practice of Error Control Codes

ECEN 5682 Theory and Practice of Error Control Codes ECEN 5682 Theory and Practice of Error Control Codes Introduction to Algebra University of Colorado Spring 2007 Motivation and For convolutional codes it was convenient to express the datawords and the

More information

Section IV.23. Factorizations of Polynomials over a Field

Section IV.23. Factorizations of Polynomials over a Field IV.23 Factorizations of Polynomials 1 Section IV.23. Factorizations of Polynomials over a Field Note. Our experience with classical algebra tells us that finding the zeros of a polynomial is equivalent

More information

Objective: To become acquainted with the basic concepts of cyclic codes and some aspects of encoder implementations for them.

Objective: To become acquainted with the basic concepts of cyclic codes and some aspects of encoder implementations for them. ECE 7670 Lecture 5 Cyclic codes Objective: To become acquainted with the basic concepts of cyclic codes and some aspects of encoder implementations for them. Reading: Chapter 5. 1 Cyclic codes Definition

More information

Introduction to finite fields

Introduction to finite fields Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in

More information

Arrangements, matroids and codes

Arrangements, matroids and codes Arrangements, matroids and codes first lecture Ruud Pellikaan joint work with Relinde Jurrius ACAGM summer school Leuven Belgium, 18 July 2011 References 2/43 1. Codes, arrangements and matroids by Relinde

More information

Week 3: January 22-26, 2018

Week 3: January 22-26, 2018 EE564/CSE554: Error Correcting Codes Spring 2018 Lecturer: Viveck R. Cadambe Week 3: January 22-26, 2018 Scribe: Yu-Tse Lin Disclaimer: These notes have not been subjected to the usual scrutiny reserved

More information

Division of Trinomials by Pentanomials and Orthogonal Arrays

Division of Trinomials by Pentanomials and Orthogonal Arrays Division of Trinomials by Pentanomials and Orthogonal Arrays School of Mathematics and Statistics Carleton University daniel@math.carleton.ca Joint work with M. Dewar, L. Moura, B. Stevens and Q. Wang

More information

RINGS: SUMMARY OF MATERIAL

RINGS: SUMMARY OF MATERIAL RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered

More information

A Brief Encounter with Linear Codes

A Brief Encounter with Linear Codes Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 8-2014 A Brief Encounter with Linear Codes Brent El-Bakri Boise State University, brentelbakri@boisestate.edu

More information

Part III. Cyclic codes

Part III. Cyclic codes Part III Cyclic codes CHAPTER 3: CYCLIC CODES, CHANNEL CODING, LIST DECODING Cyclic codes are very special linear codes. They are of large interest and importance for several reasons: They posses a rich

More information

g(x) = 1 1 x = 1 + x + x2 + x 3 + is not a polynomial, since it doesn t have finite degree. g(x) is an example of a power series.

g(x) = 1 1 x = 1 + x + x2 + x 3 + is not a polynomial, since it doesn t have finite degree. g(x) is an example of a power series. 6 Polynomial Rings We introduce a class of rings called the polynomial rings, describing computation, factorization and divisibility in such rings For the case where the coefficients come from an integral

More information

Homework 7 Solutions to Selected Problems

Homework 7 Solutions to Selected Problems Homework 7 Solutions to Selected Prolems May 9, 01 1 Chapter 16, Prolem 17 Let D e an integral domain and f(x) = a n x n +... + a 0 and g(x) = m x m +... + 0 e polynomials with coecients in D, where a

More information

MATH 433 Applied Algebra Lecture 21: Linear codes (continued). Classification of groups.

MATH 433 Applied Algebra Lecture 21: Linear codes (continued). Classification of groups. MATH 433 Applied Algebra Lecture 21: Linear codes (continued). Classification of groups. Binary codes Let us assume that a message to be transmitted is in binary form. That is, it is a word in the alphabet

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x

More information

Combinatória e Teoria de Códigos Exercises from the notes. Chapter 1

Combinatória e Teoria de Códigos Exercises from the notes. Chapter 1 Combinatória e Teoria de Códigos Exercises from the notes Chapter 1 1.1. The following binary word 01111000000?001110000?00110011001010111000000000?01110 encodes a date. The encoding method used consisted

More information

Quasi-cyclic codes. Jay A. Wood. Algebra for Secure and Reliable Communications Modeling Morelia, Michoacán, Mexico October 12, 2012

Quasi-cyclic codes. Jay A. Wood. Algebra for Secure and Reliable Communications Modeling Morelia, Michoacán, Mexico October 12, 2012 Quasi-cyclic codes Jay A. Wood Department of Mathematics Western Michigan University http://homepages.wmich.edu/ jwood/ Algebra for Secure and Reliable Communications Modeling Morelia, Michoacán, Mexico

More information

Rings in Coding Theory

Rings in Coding Theory Rings in Coding Theory Steven T. Dougherty July 3, 2013 Cyclic Codes Cyclic Codes were first studied by Prange in 1957. Prange, E. Cyclic error-correcting codes in two symbols. Technical Note TN-57-103,

More information

Self-Dual Cyclic Codes

Self-Dual Cyclic Codes Self-Dual Cyclic Codes Bas Heijne November 29, 2007 Definitions Definition Let F be the finite field with two elements and n a positive integer. An [n, k] (block)-code C is a k dimensional linear subspace

More information

Skew Cyclic Codes Of Arbitrary Length

Skew Cyclic Codes Of Arbitrary Length Skew Cyclic Codes Of Arbitrary Length Irfan Siap Department of Mathematics, Adıyaman University, Adıyaman, TURKEY, isiap@adiyaman.edu.tr Taher Abualrub Department of Mathematics and Statistics, American

More information

A first step towards the skew duadic codes

A first step towards the skew duadic codes A first step towards the skew duadic codes Delphine Boucher To cite this version: Delphine Boucher. A first step towards the skew duadic codes. 2017. HAL Id: hal-01560025 https://hal.archives-ouvertes.fr/hal-01560025v2

More information

MATH3302. Coding and Cryptography. Coding Theory

MATH3302. Coding and Cryptography. Coding Theory MATH3302 Coding and Cryptography Coding Theory 2010 Contents 1 Introduction to coding theory 2 1.1 Introduction.......................................... 2 1.2 Basic definitions and assumptions..............................

More information

} has dimension = k rank A > 0 over F. For any vector b!

} has dimension = k rank A > 0 over F. For any vector b! FINAL EXAM Math 115B, UCSB, Winter 2009 - SOLUTIONS Due in SH6518 or as an email attachment at 12:00pm, March 16, 2009. You are to work on your own, and may only consult your notes, text and the class

More information

Some Open Problems on Quasi-Twisted and Related Code Constructions and Good Quaternary Codes

Some Open Problems on Quasi-Twisted and Related Code Constructions and Good Quaternary Codes Some Open Problems on Quasi-Twisted and Related Code Constructions and Good Quaternary Codes Nuh Aydin and Tsvetan Asamov Department of Mathematics Kenyon College Gambier, OH 43022 {aydinn,asamovt}@kenyon.edu

More information

Classification of Finite Fields

Classification of Finite Fields Classification of Finite Fields In these notes we use the properties of the polynomial x pd x to classify finite fields. The importance of this polynomial is explained by the following basic proposition.

More information

Polynomials. Henry Liu, 25 November 2004

Polynomials. Henry Liu, 25 November 2004 Introduction Polynomials Henry Liu, 25 November 2004 henryliu@memphis.edu This brief set of notes contains some basic ideas and the most well-known theorems about polynomials. I have not gone into deep

More information

55 Separable Extensions

55 Separable Extensions 55 Separable Extensions In 54, we established the foundations of Galois theory, but we have no handy criterion for determining whether a given field extension is Galois or not. Even in the quite simple

More information

Explicit Methods in Algebraic Number Theory

Explicit Methods in Algebraic Number Theory Explicit Methods in Algebraic Number Theory Amalia Pizarro Madariaga Instituto de Matemáticas Universidad de Valparaíso, Chile amaliapizarro@uvcl 1 Lecture 1 11 Number fields and ring of integers Algebraic

More information

11 Minimal Distance and the Parity Check Matrix

11 Minimal Distance and the Parity Check Matrix MATH32031: Coding Theory Part 12: Hamming Codes 11 Minimal Distance and the Parity Check Matrix Theorem 23 (Distance Theorem for Linear Codes) Let C be an [n, k] F q -code with parity check matrix H. Then

More information

Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman

Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 31, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Symbolic Adjunction of Roots When dealing with subfields of C it is easy to

More information

Vector spaces. EE 387, Notes 8, Handout #12

Vector spaces. EE 387, Notes 8, Handout #12 Vector spaces EE 387, Notes 8, Handout #12 A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalar-vector product satisfying these axioms: 1. (V, +) is

More information

Cyclic Redundancy Check Codes

Cyclic Redundancy Check Codes Cyclic Redundancy Check Codes Lectures No. 17 and 18 Dr. Aoife Moloney School of Electronics and Communications Dublin Institute of Technology Overview These lectures will look at the following: Cyclic

More information