February 14, TiCC TR Generalized Residue Codes and their Idempotent Generators. Bulgarian Academy of Sciences, Bulgaria and

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1 Tlurg centre for Creatve Comutng P.O. Bo Tlurg Unversty 5000 LE Tlurg, The Netherlands htt:// Emal: Coyrght S.M. Dodunekov, A. Bolov and A.J. van Zanten Feruary 14, 2011 TCC TR Generalzed Resdue Codes and ther Idemotent Generators Bulgaran Academy of Scences, Bulgara and TCC, Tlurg Unversty S.M. Dodunekov, A. Bolov and A.J. van Zanten

2 Generalzed Resdue Codes and ther Idemotent Generators

3 Astract

4 Contents

5 1. Introducton and defntons

6 Defnton 1 {

7 Theorem 2 Eamle 3 Eamle 4

8

9 2. Idemotent generators of cyclc codes Defnton 5 Theorem 6

10 Theorem 7 Proof Corollary 8

11 Eamle 9 Eamle 10 Eamle 11

12 Eamle 12 Remark 13

13 Theorem 14 Theorem 15 Eamle 16

14 Eamle 17

15

16 Eamle 18

17 Theorem 19 Proof

18 3. A matr determnng all rmtve demotent generators

19 Lemma 20 Proof

20 Lemma 21 Theorem 22 Proof

21 Eamle 23 =.

22 Eamle 24

23 Theorem 25

24 Theorem 26 Proof Corollary 27

25 Theorem 28 Proof Remark 29 Lemma 30

26 Proof Theorem 31

27 GR- nq n q GR- C t nqt H:=<q> r C q q H H C C H SU n nn P n ssu n s Ps ss P s C s K HC tn rt KGR- g P SU C nqt e c n S C n q t th M SU n m r nq r m m SU n m GR- P SU n SU n n r t P

28 th r m m m S The demotent generator of the GR-code Cnq t S U n, s gven y, r wth n c m S tgr- C SU n M SU n Let, S, e the elements of the matr M. If and k are elements of lk SU n and f The comonents of the column vector k, k comonents of The grou U n H lu s such thatlk n U n n q t modn, then S, form a ermutaton of the acts transtvely on the set of column vectors,,..,, where,,, are the t elements of S U. t r r lk lk m m lk k lu n lk nk k S l n U n S k SU n k SU n S a S s S a U n n s a S a U n a a t U n

29 k S a k S SUn SU n U n H P n h g g P m n= n n n m t- case n n q q C q r q Pr n r r r. M m S m n q r C C C C C

30 P P P P P P P P P P M n C c c c c c s s s s ss ss n q U r r H H H H H H H H H

31 H H C H C H C H C H C H C H K K H H GF th P l lh l P lh l P lh

32 H P lh l n If s an odd rme and q an odd rme ower wth q, and f r ord q, then ord q r, for any nteger. r q k k r r q k kl q r r q q l rr. r r q r r r r n n q q q P C r P P S r q q r r M

33 S S r P P P th P GR C t q t P GR C M r M m S m GRC qt r GR C qt r q GFq S S C C q q r q r r C qq S r r a ac S ac a a a C C C C C C s a cyclotomc coset mod for all S C C and are cyclotomc cosets mod for all q t S

34 f a C, then modulo 2, ac the cyclotomc cosets n together wth the cyclotomc cosets n consttute the comlete famly of cyclotomc cosets mod and ther ndces consttute the nde set S the cyclotomc cosets n corresond to rreducle olynomals contaned n and those n to rreducle olynomals contaned n f P s an rreducle factor of, then P s an rreducle factor of and vce versa, for all S and P P f C, S and odd, s the set of ntegers a C for a even, and a for a odd, then the famly C S s the same as the famly C S as defned n C and hence P P for all Sf and only f C C a aq aq a rc a C aq aq k a q k a q a aaq C k k aq k a q CC r C C th th C C r P P C C C C P P C C C C.

35 C S S S P P P S P P P S P r P P th c P n c S n c c C C c SC S c c c c n n

36 C The demotent generator of the code q t, SU determned y theeresson r c If s the demotent corresondng to column of the matr then If s the demotent generator of the GR-code, then C qt. M for, s n, n q r C C C C P P P P M M t C P C c c c c

37 C C GRC P M M M M GF n n q q Let e a rme and q e some rme ower wth q. Let furthermore a e some ostve nteger. For odd and a or for even and a, the nequalty ord aq ord aq mles ord q ord qfor all a a q q r q r q r q k a k q q q r kk r k k r q k k k

38 r l q k k l l l k k k l l l l r k k q r e r eyy. q e r q y q q r q r a a r q q q r q r q k k r r q k k r q k k k k k r e q r e y e r q y q r q a

39 q a r q qu q n n n u u u GFq P P P r r FF F r r q r u r r q q r u r u n n F P GFq P P P

40 S P t S S q GFq P P S r r r C qt t q q q q q ord q M P S GR C qt q M P S S GFq q. Let condton hold. If r C q q s a cyclotomc coset mod wth resect to q and of sze r, then r C q q s a cyclotomc coset mod, also of sze r, whle t s a cyclotomc coset mod of sze r, for, f one relaces ntegers whch occur more than once y ust one of these.

41 C S r r P r C P r a S a S S S S a a a S S S S U a S U Let a e some nteger wth a for odd, and a for even. Let a S U a. Let, S U, e the demotent generator of the code C, qt and let a a e the demotent generator of the code C a, S U a. If a ord ord q, then. a q a a qt

42 a r n k ck m n k k k r k k m m k k k k m r n u r n k k nm k u m k M k k c c k k a- P C C a qt P a q t n qu r C C C C C C C GF GF k k ks k

43 P P P P P P P m m m m m m M M c c c c c c C C C n q r t C C C C C S P

44 P P P P P P P P P GR P P C C S M M c c c c c c c c C C n q r C C C C C C C

45 P P P P P P P r C C GR M c c c c c c c c C C n r q qu q u u n P S, S GFq P S

46 r P S GR C qt t M GR q S S r C qq S r r C a ac C S ac a a a CC CC S C S C s a cyclotomc coset mod for any S and C arecyclotomc cosets mod for all S C f a C, then mod, a C the cyclotomccosets n andtogether consttute the comlete famly of cyclotomc cosets mod and ther ndces consttute the nde set S the cyclotomc cosets ncorresond to rreducle olynomals contaned n and those n to rreducle olynomals contaned n f P s an rreducle factor of, then P s an rreducle factor of and vce versa, for all S and C f, S and odd, s the set of ntegers odd, then the famly C P r P S s the same as the famly a C for a even and C C C, and hence P P, for all S, f and only f a for a S defned n C. M n n

47 S r P S P S P r P P P r P th P P C M GR C qt n r r n q u r r GR C C C C C C C C C C C

48 C M GF P P P P P P M M C c c c c n q GF C C C C C P P P P P

49 C C C C C C C C C C th GF P P P P P P P P P P np S P S P m GRC M m m T T C C c cc c c c c c c c c c M

50 M th M C C C th C C C GR a a Let e an odd rme and q some odd rme ower wth q. Let furthermore a e some nteger. The nequalty ord q ord q mles ord q ord q, for all. a a a a a S a a S a S S S a U a Let a e some nteger satsfyng a. Let generator of the GR-code C qt S U a, S U, e the demotent a, and let e the demotent generator of C, a qt

51 . If ord q ord q a, then. a S U a a a a n q GR-C GR- C n qc C C C C C S S S U SU u ugf th P P P P P P M M

52 a n = and n = 2 n q r r r a r r r C C GF P P GR- C C C C C C C C P P P P P P P M

53 M P P P P P GR- C n q C C C C C CC CC CC C C C C C P P P P P P P P P P P P r t M GR-C

54 q q q q Let e an artrary rme and q some rme ower such that q. Let furthermore r ord q and, for. Then the followng statements hold: r f s odd, there s an nteger d, d, wth r rd rd rd, r r for d,andd d d for d f = 2, there s an nteger d, d, such that ether r r rd rd rd and d d d or r r rd rd rd and d d d, whle r r for d, and for d and d resectvely there are nonzero cyclotomc cosets mod and of sze r for r to each cyclotomc coset C q q mod r corresonds a cyclotomc coset C q q mod r to each cyclotomc coset c C q q mod corresonds a cyclotomc coset C q q of sze r wth, there of sze r, for of sze rc, rc r, there r c mod, also of sze r c, for r r r d r r d GFq r r d P P d P d P d P Sd n M

55 d M GR- C t n qt GFq r M P r P P t t If, then one has for the nteger d n Lemma 57 d for odd and d and for and q l, l odd d and for ql, l even d and for and q l, l odd d and for and q l, l even. C qt d d

56 d r r q k q lk l q l l r qll r r r r r r qll r r r r d qll r r r r qll r r r r r d n q r C r GF C r r P P C C C C r r r P P P P

57 C C C C C C C C r P P P P P P P P P P P P d q r r r r d C C C C C C C GF P P P P P P P P P P P P P P P P P P P P q d q r r r r r r d C C C C C C C C C C C C GF

58 th P P P P P P P P P P P P PPPP P P P P P P P P q q GR-GF GR- C M C t n q C C C C C C C C C

59 P P P P P P P P P GF s S M s M g PP e scs ss M M T g P Pe M M T e e e e M M T M M T s n s

60 M M T M M T M Generalzed Resdue Codes Introducton to Fnte Felds and ther Alcatons Codng Theory Generalzed Quadratc Resdue Codes The Theory of Error-Correctng Codes

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