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Lecture 19 The Vector Space Z n p (Section 5.1)

Lecture 19 The Vector Space Z n p (Section 5.1). Theory of Information. About Linear Codes. Linear codes --- most important and widely studied types of codes. Advantages of linear codes over other types of codes: Easier to describe Nearest neighbor decoding easier to implement

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Lecture 19 The Vector Space Z n p (Section 5.1)

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  1. Lecture 19The Vector Space Znp(Section 5.1) Theory of Information

  2. About Linear Codes • Linear codes --- most important and widely studied types of codes. • Advantages of linear codes over other types of codes: • Easier to describe • Nearest neighbor decoding easier to implement • Encoding and decoding messages easier • Code alphabet is a field, so code symbols can be added and multiplied • Etc. • Our attention will focus on the fields Zp, where p is prime. • Remember that Zp={0,1,…,p-1}, and • Znp is the set of all strings over Zp of length n. • Remember everything else from Lecture 3.

  3. Addition and Scalar Multiplication in Znp Assume u=u1u2…unZnp, v=v1v2…vnZnp, and Zp. Addition: u+v = u1u2…un + v1v2…vn = (u1+v1)(u2+v2)…(un+vn) Scalar multiplication: u =   (u1u2…un) = (u1)(u2)…(un) Example 5.1.1 In Z43, 1201 + 1212 = 2  (1212) = Theorem 5.1.1 If p is a prime number then Znp together with the above two operations is a vector space over Zp.

  4. Vector Spaces Definition Let F be a field. A vector space over F is a nonempty set V together with two functions: + (addition) of type VVV and  (scalar multiplication) of type FVV such that these two operations satisfy the following conditions: Associativity: For allx,y,zV, x+(y+z)=(x+y)+z Commutativity: For allx,yV, x+y=y+x Zero vector property:There exists an element 0V, called the zero vector, such that, for all xV, 0+x=x+0=x Property of negatives: For any xV, there exists another element of V, denoted by –x and calledthe negative of x, such that(-x)+x=0 Scalar multiplication property:For all x,yV and ,F, (x+y) = x+y (+)x = x+x ()x = (x) 1x = x The elements of V are called vectors, and the elements of F are called scalars. F is called the base field of V.

  5. Subspaces of Znp Definition A nonempty subset S of Znp is called a subspace of Znp if the set S, together with the operations of addition and scalar multiplication inherited from Znp, is itself a vector space. Theorem 5.1.2 A nonempty subset S of Znp is a subspace of Znp if and only if it is closed under addition and scalar multiplication. I.e., for all vectors x,yS and every scalar Zp, we have x+yS and xS. • Two observations: • A subspace S of Znp will always contains 0. Because 0x=0 must be in it. • All nonempty subsets of Zn2 are closed under scalar multiplication.

  6. Examples on Subspaces (See the textbook) Example 5.1.2 {0000,0100,0010,0110} is a subspace of Z42. Example 5.1.3 {0000,0001,1000,0110} is not a subspace of Z42. Example 5.1.4 {000,100,200,001,002,101,102,201,202} is a subspace of Z33. Example 5.1.5 {0} and Znp are subspaces of Znp.

  7. Linear Combinations Definition Let x1,x2,…xk be strings in Znp, and let 1, 2,…, k be scalars (i.e. elements of Zp). The string 1x1+ 2x2…+ kxk is called a linear combination of the strings x1,x2,…,xk, with coefficients 1, 2,…, k. If all of the coefficients are 0, then the linear combination is said to be trivial, otherwise it is nontrivial. Example 5.1.6 In Z35, the string x=2(010)+121+3(140)=411 is a (nontrivial) linear combination of 010, 121 and 140, with coefficients 2,1 and 3. The elements of the subspace S1={000,100,200,001,002,101,102,201,202} of Z33 (Example 5.1.4) can be described in terms linear combinations as S1={(100)+(001) | ,Z3}. Because of this, we say that {100,001} generates S1.

  8. Generating Sets Definition Let S be a subspace of Znp. A subset G={g1,…,gk} of S is called a generating set for S if, for any xS, there exist scalars 1,…,k such that x = 1g1 +… + kgk. To indicate that G is a generating set for S, we write S=<G>. Example 5.1.7 {0000,0100,0010,0110}=<0100,0010> (see the textbook). Theorem 5.1.3 Let G be a nonempty set of strings in Znp. The set of all linear combinations of strings in G is a subspace of Znp, called the subspace generated by G, or the subspace spanned by G, and denoted by <G>. Exercise: See that both G1={100,001} and G2={100,001,101} are generating sets for S={000,100,001,101}. G2, however, is a wasteful way to describe S, because it is not minimal.

  9. Bases • Definition Let S be a subspace of Znp. A generating set B of S is said to be a basis • for S if it is a minimal generating set, in the sense that no proper subset of B • generates S. • Example 5.1.8 B={100,001} is a basis for S={000,100,001,101}. Indeed: • B does generate S, because 000 is the trivial combination (i.e. 0(100)+0(001)); • 100=1(100) (i.e. 1(100)+0(001)); • 001=1(001) (i.e. 0(100)+1(001); • 101=100+001 (i.e. 1(100)+1(001)). • No proper subset of B can generate S. Say, {100} does not generate S because • 001 is not a scalar multiple of 100. Similarly for {001}. • We are going to see that bases allow us to describe subspaces (=linear codes) without • listing all of the elements.

  10. Standard Bases; Dimension Example 5.1.9 The set B={e1,e2,e3,e4}={1000,0100,0010,0001} is a basis for Z42. Because any string x = u1u2u3u4 in Z42 can be written as x = u1(1000) + u2(0100) + u3(0010) + u4(0001). More generally: Let ei denote the string in Znp with 0s in every position except the ith position, where it has a 1. Then B={e1,…,en} is a basis for Znp. It is called the standard basis of Znp. Theorem 5.1.6 Let S be a subspace of Znp. Then all bases for S have the same size. This size is called the dimension of S and is denoted by dim(S).

  11. Homework Exercises 3, 6, 8, 9, 12, 13. Remember (and understand) all of the items of this lecture marked as Definition or Theorem.

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