# MSS 317 Coding Theory [Teori Pengekodan]

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1 UNIVERSITI SAINS MALAYSIA First Semester Examination Academic Session 2015/2016 January 2016 MSS 31 Coding Theory [Teori Pengekodan] Duration : 3 hours [Masa : 3 jam] Please check that this examination paper consists of FIVE pages of printed materials before you begin the examination [Sila pastikan bahawa kertas peperiksaan ini mengandungi LIMA muka surat yang bercetak sebelum anda memulakan peperiksaan ini] Instructions: Answer FIVE (5) questions [Arahan: Jawab LIMA (5) soalan] In the event of any discrepancies, the English version shall be used [Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai] 2/-

2 - 2 - Answer ALL questions Jawab SEMUA soalan [ 100 markah/marks ] 1 Let C 000,001,110 and suppose that the codewords in C are sent across a noisy channel with probability of error probability is p = 01 3 For each word w F 2 that could be received, find the decoded word v C which the Maximum Likelihood Decoding Rule (MLDR) will conclude was sent [ 16 marks ] For which received words will the retransmission be requested? If v = 110 is sent, find the probability that the MLDR will correctly conclude that this v was sent [ 3 marks ] If v = 110 is sent, find the probability that the MLDR will conclude that 000 was sent 1 Biar C 000,001,110 dan andaikan bahawa semua kata kod dalam C akan dihantar melalui suatu saluran bising dengan kebarangkalian ralat ialah p = 01 3 Bagi setiap kata w F 2 yang mungkin diterima, cari kata dinyahkod v C yang akan disimpul oleh Peraturan Penyahkodan Kemungkinan Maksimum (PPKM) sebagai kata yang dihantar [ 16 markah ] Tentukan semua kata diterima yang perlukan penghantaran semula kata kodnya? Jika v = 110 dihantar, cari kebarangkalian bahawa PPKM akan menyimpul dengan betul bahawa v adalah katakod yang dihantar [ 3 markah ] Jika v =110 dihantar, cari kebarangkalian PPKM akan menyimpul bahawa 000 adalah katakod yang dihantar 3/-

3 - 3-2 Let C be the linear code with generating matrix Find a parity check matrix for C G [ 4 marks ] List the cosets of C [ 8 marks ] Use parts and to construct the syndrome look-up table for C [ 8 marks ] The matrix G was used to encode an English message according to the following key: space A E I T D H M The following message is received over a noisy channel Assuming that at most one error occurred in each word, determine the most likely intended message Do you think that the obtained message is correct? Explain your answer [ 13 marks ] 2 Biar C ialah suatu kod linear dengan matriks penjana Carikan suatu matriks semakan parity bagi C Senaraikan semua koset bagi C G [ 4 markah ] [ 8 markah ] Dengan menggunakan bahagian dan, bina satu jadual pencarian sindrom bagi C [ 8 markah ] 4/-

4 - 4 - Matriks G telah digunakan untuk mengekodkan suatu mesej bahasa English mengikut kata-kata kunci berikut: space A E I T D H M Kata-kata berikut telah diterima melalui suatu saluran bising Katakan paling banyak cuma satu ralat berlaku dalam setiap kata yang diterima, Tentukan mesej-mesej yang paling mungkin dihantar Adakah anda yakin bahawa semua mesej yang diperolehi adalah betul? Jelaskan jawapan anda [ 13 markah ] 3 Factor x 1 into irreducible polynomials over F 2 How many cyclic codes of length have 8 codewords? Explain your answer [ 5 marks ] Determine the largest size among all proper cyclic code of length where a proper cyclic code of length is a code with size m for 2 m F 2 Justify your answer [ 3 marks ] 3 Faktorkan x 1 menjadi pendaraban polynomial-polinomial yang tak terturn atas F 2 Berapa kod kitaran berpanjang yang mempunyai lapan katakod? Jelaskan jawapan anda [ 5 markah ] Tentukan saiz kod kitaran yang terbesar di kalangan semua kod kitaran berpanjang yang wajar Suatu kod kitaran berpanjang dikatakan wajar jika ia ialah suatu kod yang bersaiz size m dengan 2 m F 2 Jelaskan jawapan anda [ 3 markah ] 5/-

5 4 Construct the finite field F Construct a binary code using the finite field constructed in part a) Justify your answer by giving the three important parameters of the constructed code [ 4 marks ] 4 Bina satu medan terhingga F 8 Bina suatu kod binari dengan menggunakan medan terhingga yang dibina di bahagian Jelaskan jawapan anda dengan memberi ketiga-tiga parameter utama bagi kod tersebut [ 4 markah ] 5 Let C 1 be the [8,, 2] even parity check code and C 2 be the [8,4,4] extended binary Hamming code Define C v v u vc, u C 1 2 Show that C is a [16,11,4]-code Using a suitable bound, show that C is an optimal code Give a decoding algorithm of the code C that enables the code C to be a single error correcting and double errors detecting code Justify the mechanisms of your decoding algorithm [ 12 marks ] 5 Biar C 1 ialah suatu kod semakan parity genap yang berparameter [8,, 2] dan C 2 ialah kod binari Hamming dipanjangkan yang berparameter [8,4,4] Takrifkan C v v u vc, u C 1 2 Tunjukkan C ialah suatu kod yang berparameter [16,11,4] Dengan menggunakan suatu had yang sesuai, tunjukkan C adalah suatu kod optimal Berikan suatu algoritma penyahkodan bagi C yang dapat membetulkan satu ralat dan mengesan dua ralat Jelaskan mekanisma algoritma penyahkodan [ 12 markah ] - ooo 0 ooo -

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