MinT - Best Known (155−15, 155, s)-Nets in Base 8

Best Known (155−15, 155, s)-Nets in Base 8

(155−15, 155, 2571515)-Net over F₈ — Constructive and digital

Digital (140, 155, 2571515)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (34, 41, 174773)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 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 (31, 38, 174764)-net over F₈, using
      - net defined by OOA [i] based on linear OOA(8³⁸, 174764, F₈, 7, 7) (dual of [(174764, 7), 1223310, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(8³⁸, 524293, F₈, 7) (dual of [524293, 524255, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(8³⁸, 524294, F₈, 7) (dual of [524294, 524256, 8]-code), using
        trace code [i] based on linear OA(64¹⁹, 262147, F₆₄, 7) (dual of [262147, 262128, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(64¹⁹, 262144, F₆₄, 7) (dual of [262144, 262125, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(64¹⁶, 262144, F₆₄, 6) (dual of [262144, 262128, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (99, 114, 2396742)-net over F₈, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(155−15, 155, large)-Net over F₈ — Digital

Digital (140, 155, large)-net over F₈, using

t-expansion [i] based on digital (138, 155, large)-net over F₈, using
- 6 times m-reduction [i] based on digital (138, 161, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶¹, large, F₈, 23) (dual of [large, large−161, 24]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(155−15, 155, large)-Net in Base 8 — Upper bound on s

There is no (140, 155, large)-net in base 8, because

13 times m-reduction [i] would yield (140, 142, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (155−15, 155, s)-Nets in Base 8

(155−15, 155, 2571515)-Net over F8 — Constructive and digital

(155−15, 155, large)-Net over F8 — Digital

(155−15, 155, large)-Net in Base 8 — Upper bound on s

(155−15, 155, 2571515)-Net over F₈ — Constructive and digital

(155−15, 155, large)-Net over F₈ — Digital