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Let the primitive polynomial with degree 5 be p x x5 x2 1, and also a be a root of p x , meaning a primitive element of GF 25 . Find the minimal polynomials of a3 , a5 , and a7 . Let a be a root of primitive polynomial x2 x 2 over GF 3 . Find the minimal polynomials of all elements in GF 32 . 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Show that rows or columns of nonsingular matrix are linearly independent. And show that any nonsingular matrix can be transformed into identity matrix by elementary row operations. Construct the vector space of all 3-tuples over GF 3 . Form a two-dimensional subspace and its null space. Given the following H matrix of a linear code C, answer the questions below: 1 H 0 (a) (b) (c) (d) 1 a 1 b b a ! over GF 4 :PDF 417 integration with .netuse visual studio .net pdf 417 encoder togenerate pdf-417 2d barcode in .net2.14 2.15Codeabar implement for .netusing barcode creation for visual studio .net crystal control to generate, create codabar image in visual studio .net crystal applications.Transform the above matrix to a systematic form. Find all codewords of C. Encode a given message a b . Decode a received word 1 b b 0 .UPC - 13 barcode library for visual c#use .net vs 2010 ean13 creation toget european article number 13 with visual c#.netProve that the code is capable of correcting any combinations of random t errors and e erasures if the minimum distance of the code dmin is at least 2t e 1.Rdlc Report upc - 13 printeron .netusing barcode maker for rdlc reports net control to generate, create ean13+5 image in rdlc reports net applications.BASIC MATRIX CODES Web Form Crystal ucc - 12 writeron .net c#using barcode maker for asp.net web pages crystal control to generate, create ucc - 12 image in asp.net web pages crystal applications.Let a linear code be expressed by the parity-check matrix H, 1 H 1 1 a 1 a2 1 0 0 1 ! over GF 4 ;Control ucc - 12 image on visual basic.netuse visual studio .net implementation toembed on visual basicwhere a is a root of the binary polynomial x2 x 1. (a) (b) (c) (d) (e) Show that the set f0; 1; a; a2 g is GF 4 . Show that the code is a single-error correcting code. For the given input information 1 0 a2 , nd the transmitted word. Decode the received word 1 a 0 a2 1 . For the single-error correcting code over GF 4 expressed by a parity-check matrix with r rows, express the maximum code length by using r. Also mention how the answer is deduced.UPC-A barcode library in javausing barcode creator for java control to generate, create ucc - 12 image in java applications.Let C be a binary code with code length n and information length k whose paritycheck matrix H is organized by distinct odd-weight-column vectors, each with length r n k . Here weight of a vector means the number of 1 s in a vector. (a) Construct the parity-check matrix H of code C for n 16 and k 11. (b) Express the maximum code length of code C by using r. (c) Prove that sum of arbitrary two column vectors in H is an even-weight column vector. (d) Prove that the code C with maximum code length has the minimum Hamming distance 4. (e) Prove that every codeword of C has even weight.Bar Code barcode library in .netuse .net winforms bar code implementation toconnect bar code on .net2.19 2.20IReport bar code generatingon javausing barcode drawer for ireport control to generate, create barcode image in ireport applications.If an error pattern e x is detectable for a cyclic code, show that its cyclic shifted pattern of e x is also detectable. Let g x be a generator polynomial with degree n k of the n; k cyclic code C. Prove that g x has the following properties: 1. g x is a monic polynomial with constant term equal to 1, and is uniquely determined. 2. g x is a factor of xn 1, meaning xn 1 g x h x , where h x , called a parity-check polynomial, has degree k. 3. The reciprocal of h x , de ned as xk h 1=x , is also a factor of xn 1, and generates a cyclic code, that is, a dual code of C.Control ucc - 12 data for visual basic gtin - 128 data with visual basic2.21 2.22 2.23For the 7; 4 cyclic code C de ned by the generator polynomial g x x3 x 1, show that the dual code of C has minimum Hamming distance 4. Given a 7; 4 cyclic SEC code C with a generator polynomial g x x3 x 1, obtain the dual code of C which has an SEC-DED capability. Prove Theorem 2.5.