MinT - Best Known (153−17, 153, s)-Nets in Base 8

Best Known (153−17, 153, s)-Nets in Base 8

(153−17, 153, 2097358)-Net over F₈ — Constructive and digital

Digital (136, 153, 2097358)-net over F₈, using

8¹ times duplication [i] based on digital (135, 152, 2097358)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 22, 208)-net over F₈, using
    - trace code for nets [i] based on digital (3, 11, 104)-net over F₆₄, using
      - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
  2. digital (113, 130, 2097150)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8¹³⁰, 2097150, F₈, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹³⁰, 8388601, F₈, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(8¹³⁰, 8388602, F₈, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
        trace code [i] based on linear OOA(64⁶⁵, 4194301, F₆₄, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶⁵, 8388602, F₆₄, 17) (dual of [8388602, 8388537, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(153−17, 153, 2097664)-Net in Base 8 — Constructive

(136, 153, 2097664)-net in base 8, using

8¹ times duplication [i] based on (135, 152, 2097664)-net in base 8, using
- base change [i] based on digital (97, 114, 2097664)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (8, 16, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 8, 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]
    2. digital (81, 98, 2097150)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16⁹⁸, 2097150, F₁₆, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
        OOA 4-folding and stacking with additional row [i] based on linear OOA(16⁹⁸, 8388601, F₁₆, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁹⁸, 8388602, F₁₆, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
        trace code [i] based on linear OOA(256⁴⁹, 4194301, F₂₅₆, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256⁴⁹, 8388602, F₂₅₆, 17) (dual of [8388602, 8388553, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁹, large, F₂₅₆, 17) (dual of [large, large−49, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(153−17, 153, large)-Net over F₈ — Digital

Digital (136, 153, large)-net over F₈, using

t-expansion [i] based on digital (131, 153, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁵³, large, F₈, 22) (dual of [large, large−153, 23]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(153−17, 153, large)-Net in Base 8 — Upper bound on s

There is no (136, 153, large)-net in base 8, because

15 times m-reduction [i] would yield (136, 138, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (153−17, 153, s)-Nets in Base 8

(153−17, 153, 2097358)-Net over F8 — Constructive and digital

(153−17, 153, 2097664)-Net in Base 8 — Constructive

(153−17, 153, large)-Net over F8 — Digital

(153−17, 153, large)-Net in Base 8 — Upper bound on s

(153−17, 153, 2097358)-Net over F₈ — Constructive and digital

(153−17, 153, large)-Net over F₈ — Digital