MinT - Best Known (20−5, 20, s)-Nets in Base 8

Best Known (20−5, 20, s)-Nets in Base 8

(20−5, 20, 12098)-Net over F₈ — Constructive and digital

Digital (15, 20, 12098)-net over F₈, using

net defined by OOA [i] based on linear OOA(8²⁰, 12098, F₈, 6, 5) (dual of [(12098, 6), 72568, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8²⁰, 12099, F₈, 2, 5) (dual of [(12099, 2), 24178, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(8¹, 4033, F₈, 2, 1) (dual of [(4033, 2), 8065, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹, s, F₈, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    2. linear OOA(8⁵, 4033, F₈, 2, 2) (dual of [(4033, 2), 8061, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁵, 4681, F₈, 2, 2) (dual of [(4681, 2), 9357, 3]-NRT-code), using
        appending kth column [i] based on linear OA(8⁵, 4681, F₈, 2) (dual of [4681, 4676, 3]-code), using
        Hamming code H(5,8) [i]
    3. 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]

(20−5, 20, 17428)-Net over F₈ — Digital

Digital (15, 20, 17428)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8²⁰, 17428, F₈, 5) (dual of [17428, 17408, 6]-code), using
- (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
  1. linear OA(8¹, 4681, F₈, 1) (dual of [4681, 4680, 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]
  2. linear OA(8⁵, 4681, F₈, 2) (dual of [4681, 4676, 3]-code), using
    - Hamming code H(5,8) [i]
  3. 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]

(20−5, 20, 65281)-Net in Base 8 — Constructive

(15, 20, 65281)-net in base 8, using

base change [i] based on digital (10, 15, 65281)-net over F₁₆, using
- net defined by OOA [i] based on linear OOA(16¹⁵, 65281, F₁₆, 5, 5) (dual of [(65281, 5), 326390, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(16¹⁵, 130563, F₁₆, 5) (dual of [130563, 130548, 6]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(16¹⁴, 130562, F₁₆, 5) (dual of [130562, 130548, 6]-code), using
      - trace code [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]

(20−5, 20, large)-Net in Base 8 — Upper bound on s

There is no (15, 20, large)-net in base 8, because

3 times m-reduction [i] would yield (15, 17, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]