MinT - Best Known (65−17, 65, s)-Nets in Base 8

Best Known (65−17, 65, s)-Nets in Base 8

(65−17, 65, 521)-Net over F₈ — Constructive and digital

Digital (48, 65, 521)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (40, 57, 512)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵⁷, 512, F₈, 17, 17) (dual of [(512, 17), 8647, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(8⁵⁷, 4097, F₈, 17) (dual of [4097, 4040, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(65−17, 65, 674)-Net in Base 8 — Constructive

(48, 65, 674)-net in base 8, using

8¹ times duplication [i] based on (47, 64, 674)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 18, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 9, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. (29, 46, 514)-net in base 8, using
    - trace code for nets [i] based on (6, 23, 257)-net in base 64, using
      - 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
        base change [i] based on digital (0, 18, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(65−17, 65, 4682)-Net over F₈ — Digital

Digital (48, 65, 4682)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁵, 4682, F₈, 17) (dual of [4682, 4617, 18]-code), using
- 574 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 7 times 0, 1, 22 times 0, 1, 61 times 0, 1, 154 times 0, 1, 323 times 0) [i] based on linear OA(8⁵⁸, 4101, F₈, 17) (dual of [4101, 4043, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(14) [i] based on
    1. linear OA(8⁵⁷, 4096, F₈, 17) (dual of [4096, 4039, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(8⁵³, 4096, F₈, 15) (dual of [4096, 4043, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    3. linear OA(8¹, 5, F₈, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(65−17, 65, large)-Net in Base 8 — Upper bound on s

There is no (48, 65, large)-net in base 8, because

15 times m-reduction [i] would yield (48, 50, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]