CONSTACYCLIC CODES OF LENGTH OVER A FINITE FIELD
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1 Jorl o Algbr Nbr Tory: Ac Alco Vol 5 Nbr 6 Pg 4-64 Albl ://ccc.co. DOI: ://.o.org/.864/_753 ONSTAYLI ODES OF LENGTH OVER A FINITE FIELD AITA SAHNI POONA TRAA SEHGAL r or Ac Sy c Pb Ury gr 64 I -l: 5@gl.co oo.gl@gl.co Abrc L F b l w l wr o r owr. W lcly r λ -cocyclc co o lg r l r o gror olyol wr or F r c o r cor o. All r roo cocyclc co o lg or F r l r lo crcrz r o gror olyol. T br o c co lo b o.. Iroco Sc 95 cyclc gcyclc co b ly. T co c b ob ro cl o cocyclc co. ocyclc co or or cl o lr co o c Sbc lco: T7 94B5 94B5. Kywor r: cocyclc co gror olyol l co cyclooc co. Rc Jly Scc Ac Pblr
2 4 AITA SAHNI POONA TRAA SEHGAL r rc lgbrc rcr wc r l or c rror co corrco. L F o l w l. A [ ] N lr co o lg N o or F N r cll cowor r wr row cor: c ( c c -ol bc o cor c F. El o bc c N ). A lr co or F cll λ-cocyclc co ( λc N N c c N ) or ry ( c c c N ). L θ : F F [ ] N b g by θ(( c c cn )) c c λ > N N c N o( λ ). O c ly cc θ F -ol oor. W c ror y λ-cocyclc co o lg N F [ ] or F w l. N I λ co r cyclc λ > co or λ co r ow gcyclc co. I N cor o crcrc o F λ-cocyclc co o lg N or F cll l-roo λ-cocyclc co orw cll r roo λ-cocyclc co. Sl roo λ-cocyclc co o g lg or l b ly. I [] B R l roo µ-cocyclc co o lg or l o µ owr o. Ty rl o r roo λ-cocyclc co o lg oor ro F F bg o r owr w orr or F or y o zro λ F [ ] λ > o F [ ] wr orr o Λ > Λ g owr o occrrg orr o λ crcrc o F. I [4] l. crcrz olyol gror o cocyclc co o lg wr crcrc o l r r ro. I [5 6 7] D ob cocyclc co o lg 3 or l F. 4
3 ONSTAYLI ODES OF LENGTH 43 I ro wor [8] or l-roo cyclc co o lg or l. I [9] l-l lorogol l-roo gcyclc co or l b. Hr l-orogol cocyclc co or l o DS Hr l-orogol cocyclc co r ob []. I r w y λ-cocyclc co o lg or l F wr r c o r o r owr cor o c r roo olo ( ). I Sco gror olyol br o co r o. I Sco 3 w cor l co o cocyclc co. Elc ro or gror olyol o corrog l co r g. I Sco 4 r roo cocyclc co o lg l r r. or l F r. Sl Roo ocyclc o o Lg L b c o r b o r owr r roo olo ( ). L λ b o zro l o F. T orr o λ o by o ( λ ) r ( ). I co w cor l roo λ-cocyclc co o lg F or. or L. For r bo o o r. Proo. I r c r ( ) w ( ) wc corco r roo olo o r.
4 44 AITA SAHNI POONA TRAA SEHGAL Wr r w w. L. Tr Λ β F w o ( Λ ) orr o β o bl by c λ Λβ. Proo. Wr c > c. Obr o( λ ) w or o ( ). Tror w c. A cor r gr b c c b c r c b b o λ λ λ ( λ ) Λβ c wr c Λ λ b β λ. A o( λ ) w β orr w wc o bl by. Alo c c w Λ λ λ w c w c Λ λ (c c r cor o ). T o ( Λ ). To ro o Λ β o r or Λ β yg λ Λ β wr o ( Λ ) orr o β o bl by. Tror Λ β Λ β o ΛΛ β β. B ΛΛ orr owr o β β orr cor o. T ΛΛ β β gg Λ Λ β β. Alo β β b b c b β c b ( β ) ( β ) β. T ro l.
5 ONSTAYLI ODES OF LENGTH 45 L 3. L λ Λ β b L. T Φ : [ ] [ ] F F λ > Λ > g by ( β) rg oor. Proo. lrly Φ rg ooor oo. L g F [ ] b c Φ ( ) Φ( g ). T ( β ) g( β) o( Λ ).. g ( β β Λ ) or o F [ ]. Rlcg by β w g g ( β Λ ) ( β ) β β ( Λβ ) ( β ) ( λ ) ( β ) o( λ ). T g o ( λ ) gg Φ o o rr. L 4. > Λ-cocyclc co o lg or F oly ( β ) > λ-cocyclc co o lg or F. Proo. Follow ro oor L 3. lg I w o L 4 og o cor Λ-cocyclc co o or F wr o ( Λ ) or o. L 5. ( Λ ) ( ). Proo. L α b roo o ( Λ ). T α Λ orr o Λ gg ( α ) α Λ. Howr
6 46 AITA SAHNI POONA TRAA SEHGAL ( Λ ) l roo or F ollow ( Λ ) ( ). I orr o gror olyol o Λ-cocyclc co w o corzo o ( Λ ). For y N l { } o -cyclooc co olo cog wr N ( o ). L α b r N l o gr c - roo o y o o l o F. W ow ( α ) l olyol o α or F wr r or o rr o ll c -cyclooc co olo. L b gr c ( ). L 6. L b. T orr o olo { } ( ). Proo. r roo olo o orr o olo ( ). Alo c.. ( o ). Tror orr o olo { }. By L [8] orr o olo lc[ ( ) { } ]. A ( ) lc[ ( ) { } ] { } ( ).
7 ONSTAYLI ODES OF LENGTH 47 L 7. Tr gr gc c r roo olo orr o olo. ϕ Proo. Sc gc r gr c. L g b r roo olo. Alo r roo olo. L b c. o g Obr g o g g o o r roo olo. Slrly o o orr o olo. Tor. L b b gr L 7. T cyclooc - co olo r g by { } or γ γ or θ θ wr { } γ { }. θ
8 AITA SAHNI POONA TRAA SEHGAL 48 Proo. By L 6 w orr o olo γ or. T γ o cyclooc co cog. γ Alo orr o olo θ o θ o cyclooc co cog. θ W ow co r rw o. For o γ l. T r γ c o... o T ( o ) o lyg. o Sc r roo olo w g ( o ) o. Wr. Now ( o ) or. o Alo r roo olo orr θ olo o orr o olo. γ Tror r γ o or o gr r. T r γ o gg ( ). o r γ Hc γ or γ owg. Slrly or or o. θ T r θ c ( o
9 ONSTAYLI ODES OF LENGTH 49 )... o Ag r roo olo orr θ olo o r θ. o T r θ o gg ( ). o r θ Hc θ or θ lyg. T cyclooc co r rw o. Flly r ll cyclooc co o bc cog crl θ γ θ γ θ γ θ θ γ γ. W coo α r - roo o y o o l o F c. Λ α T. θ γ (.)
10 5 AITA SAHNI POONA TRAA SEHGAL Tor. For y rrcbl olyol ( Λ ) oly ( ). Proo. ( Λ ) oly α roo o ( Λ ).. oly α Λ or Λ Λ wc r oly ( o ) ( o( Λ) ). L 8. ( ) ( Λ ) oly ( o ). Proo. By Tor ( ) ( Λ ) oly ( o ) lyg ( o ). L orr o olo b. Wr ( ). Alo r r roo olo o ( o ) or o gr r gc( r ). Tor 3. T corzo o ( Λ ) or F o oc rrcbl cor g by () For ( Λ ) ( ) wr or S S S ( { { } )} (.) or S ( { { } )}. (.3)
11 ONSTAYLI ODES OF LENGTH 5 () For wr or ( Λ ) S S { ; { }} (.4) S { ; { }}. (.5) () For > ( Λ ) S wr or { { } γ S r ; } o ( ) (.6) { { } θ S r ; } o ( ). (.7) Proo. () I rl ollow ro (.). () I w ( o ). T L 8 l ( Λ ) oly ( o ). Alo r roo olo γ o ( o ( )). T ( ) or c ( ) or θ c. T bo c ( Λ ) oly S.
12 5 AITA SAHNI POONA TRAA SEHGAL r () I > w ( o ) wr orr o olo. T ( o ) ( ) oly r ( o ).. r ( o ( )) or r( o ). T ( ) ( Λ ) oly S. Tor 4. L b Λ b o zro l o F w o ( Λ ). T gror olyol o Λ-cocyclc co o lg or F r g by: () For Λ-cocyclc co o lg g by or F r ( ) ( ) ( ) S S wr S (.) (.3). () For Λ-cocyclc co o lg g by wr S S (.4) (.5). ( ) () For > Λ-cocyclc co o lg g by wr S S (.6) (.7). ( ) or F r or F r Hr l or or c rl.
13 lg ONSTAYLI ODES OF LENGTH 53 Tor 5. Tr r rcly or F wr () For ζ br o λ-cocyclc co o ζ ( ) ( ) ( )( ) { } >. () For ζ ( ) ( ) >. 3. Dl o I co w g l co o Λ-cocyclc co o lg or F. L b Λ-cocyclc co o lg ( Λ ) or F gr by g. L b cc g olyol. I wll ow l co o Λ -cocyclc co o lg or F gror olyol g wr ( ) rcrocl olyol o. No oc olyol ( Λ ).
14 54 AITA SAHNI POONA TRAA SEHGAL L 9. I o -cyclooc co o cog ( γ ). Proo. L γ or o ( ). T r γ gr ( ) c ( o ). T ( o ) ( o ). Sc ( o ) w g ( o ) or ( ) ( o ). Sc r roo olo w ( ) ( o ( )) ( ).. ( o ) gg bl by. Wr. Alo ( o ).. ( o ). Now r roo olo θ orr o o γ orr o. T γ ( ) r ( o ) or o gr r. Hc γ ( ) r ( o ) or γ ( ( ) r ) ( o ) γ gg ( ) r ( ) ( o ). Tror owg or γ ( ) γ or ( ) γ ( ). I ( o ) or o. T ( o ) wr ( o ) o r gg corco. T ( γ ).
15 ONSTAYLI ODES OF LENGTH 55 L. For γ wr γ γ γ γ γ ; ; (3.) or θ wr θ θ θ θ θ. ; ; (3.) Rr. L b Λ-cocyclc co o lg or F w gror olyol ( ) g wr rl l g Tor 4. T ( ) ( ) ( ) ( ). T l co Λ -cocyclc co o lg or F w gror olyol ( ). Tor 6. L b Λ-cocyclc co o lg or F w gror olyol g Tor 4. T l co Λ -cocyclc co o lg or F w gror olyol g by ollowg:
16 56 AITA SAHNI POONA TRAA SEHGAL () For S S ( ) ( ) ( ) ; () For S S ( ) ( ) ; () For > S ( ) ( ) S wr r (3.) (3.) rcly. Proo. Follow ly ro L Rr. Now w cor λ-cocyclc co o lg or F wr o( λ ) w w. W λ-cocyclc co o lg or F r oorc o Λ-cocyclc co o lg wr λ Λβ o( Λ) r g or F Φ : [ ] [ ] F F λ > Λ > g by ( β). oror by L 4 > Λ-cocyclc co o lg or F oly ( β ) > λ-cocyclc co o lg or F. Tor 7. For y Λ-cocyclc co o lg ( Φ ( )) Φ ( ). or F
17 ONSTAYLI ODES OF LENGTH 57 Proo. L b Λ-cocyclc co o lg or F gr by g. Wr g Λ. T g( β ) ( β ) β ( λ ) o Φ ( ) gr by g( β ). T ( Φ ( )) gr by ( β ).. ( Φ ( )) ( β ) >. (3.3) O or >. By L 3 w ob Φ ( ) ( β ) >. (3.4) T rl ollow ro (3.3) (3.4). El. T T. T 83. () For Λ w 7 ( ) wr ( 3 ) 83 3 ( 3 )
18 58 AITA SAHNI POONA TRAA SEHGAL () For Λ 4. For Λ w 7 ( ) ( ) ( ) ( ) ( ) wr ( ) ( ) ( ) ( )
19 ONSTAYLI ODES OF LENGTH 59 For Λ 4 w 7 ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) wr ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) T ollowg bl g br gror olyol o λ-cocyclc co o lg 7 or F 7 :
20 λ o ( λ) Λ β ζ g 4 ( ) ( 3 ) ( ) 83 3 ( 3) ( ) ( ) 4 ( ) 4 3 ( ) ( 4) ( ) 4 ( 4) 4 ( 4 ) 4 3 ( ) 83 ( 4) ( ) 4 ( 4) 4 ( 4 ) 4 3 ( ) ( ) 4 ( ) 4 3 ( ) ( ( )) wr l or. ( 3 ) ( ( )) 83 3 ( 3) ( )
21 ONSTAYLI ODES OF LENGTH 6 4. R Roo ocyclc o o Lg or F I co w cor r roo cocyclc co o lg or F. L λ F b rbrry l w o ( λ ) w. T r Λ β F w o ( Λ ) o ( β) o bl by c λ Λβ ( L ). Alo gc ( ) o r gr b c b ( o( )).. b ( o ) or ( ). Now gc ( b ). Tror orr o Λ b l o orr o Λ F Alo b.. L. L λ Λ β b bo. T Φ : F [ ] F [ ] λ > Λ > g by b ( β ) rg oor. Proo. Slr o L 3. L. > Λ-cocyclc co o lg or F oly ( β ) > or F. b λ-cocyclc co o lg Tor 8. W oo bo l b ( Λ )
22 AITA SAHNI POONA TRAA SEHGAL 6 b corzo o b Λ o rrcbl cor or F g Tor 4. T ( ) ( ) ( ) b b b β β β c or r rcly ll r-roo λ-cocyclc co o lg or. F Proo. A r or ll or o Λ or > F r rcly ll Λ-cocyclc co o lg or. F Now b b Λ Λ Λ ( ) ( ) ( ) ( ). T ( ) ( ) ( ) r rcly ll r-roo Λ-cocyclc co o lg or. F T or ow ollow ro L. orollry. Tr r rcly ( ) ζ br o r roo λ-cocyclc co o lg or F wr () For { } > ζ.
23 ONSTAYLI ODES OF LENGTH 63 () For ζ ( ) ( ). Tor 9. L b Λ-cocyclc co o lg w gror or F T Λ -cocyclc co ( ). or F gr by ( ). by L b Λ-cocyclc co o lg or F gr g ( ). A g Λ o >. L Φ b b L Φ ( ) ( β ) >. b b Alo g( β ) ( β ) β b ( λ ) b Φ ( ) g( β ) > o ( Φ ( )) b ( β ) >. oly w ro ollowg: Tor. ( Φ ( )) Φ ( ) or ll Λ-cocyclc co o lg or F.
24 64 AITA SAHNI POONA TRAA SEHGAL Rrc [] G. K. B. R A cl o cocyclc co or l F Fl Al. 8() () [] T. Blcor Ngcyclc c co F Fl Al. 4(4) (8) [3] T. Blcor Iol cocyclc co F Fl Al. 4 (3) [4] Bocog Y F Lr L Hogw L ocyclc co or l F Fl Al. 8(6) () 7-3. [5] H. Q. D R-roo cocyclc co o lg F Fl Al. 8() () [6] H. Q. D Srcr o r-roo cocyclc co o lg l Dcr c 33 (3) r [7] H. Q. D O r-roo cocyclc co o lg 4 A-Ero J (3) [5 g]. [8] A. S P. T. Sgl l cyclc co o lg F Fl Al. 8 () [9] A. S P. T. Sgl Ero o l l l orogol gcyclc co or l Ac c o oco 9(4) (5) [] A. S P. T. Sgl Hr l-orogol cocyclc co or l Jorl o Dcr c (4) [7 g]. g
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