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EXERCISES
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Prove that the following code shows the (72, 64) S4ECjSEC code with N0 16, N1 56, L 4; and b 4: ! I I I I P P P jPj3 I O H ; I T T2 T3 T4 P T5 P T6 P jT7 Pj3 O I where T: 4 4 companion matrix de ned by g x x4 x 1, I: 4 4 identity matrix, O: 4 4 zero matrix, P: parity check matrix of (15,11) SEC code, 1 60 P 6 40 0 2 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 1 1 1 3 1 17 7; 15 1
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jPj3 : rst 3 columns in P, jT7 Pj3 : rst 3 columns in T7 P. (c) Referring to the code design shown in (b), design the generalized matrix form of the N; N R SbECjSEC code using the maximal SbEC codes indicated in Subsection 5.1.4.
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Prove that the Bl ECjSEC codes denoted by Theorem 10.12 satisfy the following relation in addition to those indicated in Theorem 10.10: r > min c m ; LCM c; 2m 1 ; 2l 1 c; l m;
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where LCM x; y means a least common multiple of x and y. 10.13 Design the 2-level l1 -bit burst error correcting and l2 -bit burst error correcting UEP codes, denoted as (Bl1 EC)n0 -(Bl2 EC)n1 UEP codes, where l1 > l2 , which can be designed by applying the interleaving method to the (Bl EC)n0 -(SEC)n1 codes. In your design of the codes, also use conversion matrix P mentioned in Subsection 10.3.2. Design the 2-level l1 -bit burst error correcting and l2 -bit burst error correcting UEC codes (called Bl1 ECjBl2 EC code) where l1 > l2 . Prove Theorem 10.17. Design an (18, 15) F2 ECjSEC code over GF(4). Design a (34, 28) (F3 S)EC code over GF(5).
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10.14 10.15 10.16 10.17
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CODES FOR UNEQUAL ERROR CONTROL / PROTECTION (UEC / UEP)
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Recall the two-level burst error control q-ary UEC code, denoted as Bl=n0 ECjSEC code, capable of correcting q-ary l-symbol burst errors in X0 with length n0 symbols as well as correcting q-ary single-symbol errors in X1 with length n1 n n0 symbols. Answer the following questions. (a) Find the necessary and suf cient conditions of this code. (b) Prove that a linear n; n r q-ary Bl=n0 ECjSEC code exists only if n qr q 1 n0 l q ql 1 n0 ; q 1
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where n indicates the code length in symbols and r the check-symbol length. (c) Show that the null space of the following matrix H over GF q is an optimal linear q-ary Bl=n0 ECjSEC code that satis es the bound on code length shown in (b): 3x 2 l2 H0;0 7 6 2 3 6 7 H0;1 H1 7 6 7 6 H0 H1 7 6 7 6 . 7 6 . H 6 7 2l ; . 4 5 6 7 7 6 00 0 11 1 7y 6 H0; l 1 5 4 0 0 0 n0 ! where 2 1 1 1 n1 n n0 ! l1
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3x 60 0 7 0 0 0 0 6 7 6 aq a1 a0 7 2 ; 0 07 0 6 40 0 5 0 0 0 0 j j j y l ! l ! l ! n0 ! j j j 2l n0 q 1 l
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H0;i : 2 n0 matrix that shifts i symbols cyclically leftward in H0;0 ; 1 i < l 1, 2 6 6 j 6 6 n H1 6 a11 2 6 6 6 4 j j an1 3 1 j j a1 1 j j a0 1 j 3x 07 7 7 07 7 ; . 7 2l .7 .7 5 0 y
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