MinT - Best Known (24−5, 24, s)-Nets in Base 8

Best Known (24−5, 24, s)-Nets in Base 8

(24−5, 24, 32335)-Net over F₈ — Constructive and digital

Digital (19, 24, 32335)-net over F₈, using

net defined by OOA [i] based on linear OOA(8²⁴, 32335, F₈, 6, 5) (dual of [(32335, 6), 193986, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8²⁴, 32336, F₈, 2, 5) (dual of [(32336, 2), 64648, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(8⁰, 4042, F₈, 2, 0) (dual of [(4042, 2), 8084, 1]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁰, s, F₈, 2, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
    2. linear OOA(8⁰, 4042, F₈, 2, 0) (dual of [(4042, 2), 8084, 1]-NRT-code) (see above)
    3. linear OOA(8⁰, 4042, F₈, 2, 0) (dual of [(4042, 2), 8084, 1]-NRT-code) (see above)
    4. linear OOA(8¹, 4042, F₈, 2, 1) (dual of [(4042, 2), 8083, 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]
    5. linear OOA(8¹, 4042, F₈, 2, 1) (dual of [(4042, 2), 8083, 2]-NRT-code) (see above)
    6. linear OOA(8¹, 4042, F₈, 2, 1) (dual of [(4042, 2), 8083, 2]-NRT-code) (see above)
    7. linear OOA(8⁵, 4042, F₈, 2, 2) (dual of [(4042, 2), 8079, 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]
    8. linear OOA(8¹⁶, 4042, F₈, 2, 5) (dual of [(4042, 2), 8068, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        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]
        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]

(24−5, 24, 69961)-Net in Base 8 — Constructive

(19, 24, 69961)-net in base 8, using

net defined by OOA [i] based on OOA(8²⁴, 69961, S₈, 6, 5), using
- OOA stacking with additional row [i] based on OOA(8²⁴, 69962, S₈, 2, 5), using
  - (u,Â u+v)-construction [i] based on
    1. 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]
    2. OOA(8¹⁹, 65281, S₈, 2, 5), using
      - OOA 2-folding [i] based on OA(8¹⁹, 130562, S₈, 5), using
        discarding parts of the base [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]

(24−5, 24, 82890)-Net over F₈ — Digital

Digital (19, 24, 82890)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(24−5, 24, 134931)-Net in Base 8

(19, 24, 134931)-net in base 8, using

base change [i] based on digital (13, 18, 134931)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁸, 134931, F₁₆, 5) (dual of [134931, 134913, 6]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(16⁴, 4369, F₁₆, 2) (dual of [4369, 4365, 3]-code), using
      - Hamming code H(4,16) [i]
    2. 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]

(24−5, 24, large)-Net in Base 8 — Upper bound on s

There is no (19, 24, large)-net in base 8, because

3 times m-reduction [i] would yield (19, 21, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]