MinT - Best Known (11, 16, s)-Nets in Base 8

Best Known (11, 16, s)-Nets in Base 8

(11, 16, 4041)-Net over F₈ — Constructive and digital

Digital (11, 16, 4041)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹⁶, 4041, F₈, 6, 5) (dual of [(4041, 6), 24230, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8¹⁶, 4042, F₈, 2, 5) (dual of [(4042, 2), 8068, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(8², 9, F₈, 2, 2) (dual of [(9, 2), 16, 3]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(2;16,8) [i]
    2. linear OOA(8¹⁴, 4033, F₈, 2, 5) (dual of [(4033, 2), 8052, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(8¹⁴, 8066, F₈, 5) (dual of [8066, 8052, 6]-code), using
        trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]

(11, 16, 8075)-Net over F₈ — Digital

Digital (11, 16, 8075)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶, 8075, F₈, 5) (dual of [8075, 8059, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(8², 9, F₈, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
    - extended Reedâ€“Solomon code RS_e(7,8) [i]
    - Hamming code H(2,8) [i]
  2. linear OA(8¹⁴, 8066, F₈, 5) (dual of [8066, 8052, 6]-code), using
    - trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
      - a code by Danev and Olsson [i]

(11, 16, 1198371)-Net in Base 8 — Upper bound on s

There is no (11, 16, 1198372)-net in base 8, because

1 times m-reduction [i] would yield (11, 15, 1198372)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 35 184384 671731 > 8¹⁵ [i]