MinT - Best Known (33−11, 33, s)-Nets in Base 8

Best Known (33−11, 33, s)-Nets in Base 8

(33−11, 33, 260)-Net over F₈ — Constructive and digital

Digital (22, 33, 260)-net over F₈, using

8¹ times duplication [i] based on digital (21, 32, 260)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 10, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 5, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (11, 22, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 11, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(33−11, 33, 516)-Net in Base 8 — Constructive

(22, 33, 516)-net in base 8, using

8¹ times duplication [i] based on (21, 32, 516)-net in base 8, using
- base change [i] based on digital (13, 24, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 12, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(33−11, 33, 650)-Net over F₈ — Digital

Digital (22, 33, 650)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8³³, 650, F₈, 11) (dual of [650, 617, 12]-code), using
- 128 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 13 times 0, 1, 34 times 0, 1, 74 times 0) [i] based on linear OA(8²⁸, 517, F₈, 11) (dual of [517, 489, 12]-code), using
  - constructionÂ XX applied to C₁Â =Â C([510,8]), C₂Â =Â C([0,9]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,8]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([510,9]) [i] based on
    1. linear OA(8²⁵, 511, F₈, 10) (dual of [511, 486, 11]-code), using the primitive BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â {−1,0,…,8}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    2. linear OA(8²⁵, 511, F₈, 10) (dual of [511, 486, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(8²⁸, 511, F₈, 11) (dual of [511, 483, 12]-code), using the primitive BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â {−1,0,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    4. linear OA(8²², 511, F₈, 9) (dual of [511, 489, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    5. linear OA(8⁰, 3, F₈, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
    6. linear OA(8⁰, 3, F₈, 0) (dual of [3, 3, 1]-code) (see above)

(33−11, 33, 224134)-Net in Base 8 — Upper bound on s

There is no (22, 33, 224135)-net in base 8, because

1 times m-reduction [i] would yield (22, 32, 224135)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 79228 903911 706696 451144 287490 > 8³² [i]

Best Known (33−11, 33, s)-Nets in Base 8

(33−11, 33, 260)-Net over F8 — Constructive and digital

(33−11, 33, 516)-Net in Base 8 — Constructive

(33−11, 33, 650)-Net over F8 — Digital

(33−11, 33, 224134)-Net in Base 8 — Upper bound on s

(33−11, 33, 260)-Net over F₈ — Constructive and digital

(33−11, 33, 650)-Net over F₈ — Digital