MinT - Best Known (22−5, 22, s)-Nets in Base 8

Best Known (22−5, 22, s)-Nets in Base 8

(22−5, 22, 32263)-Net over F₈ — Constructive and digital

Digital (17, 22, 32263)-net over F₈, using

net defined by OOA [i] based on linear OOA(8²², 32263, F₈, 6, 5) (dual of [(32263, 6), 193556, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8²², 32264, F₈, 2, 5) (dual of [(32264, 2), 64506, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(8⁰, 4033, F₈, 2, 0) (dual of [(4033, 2), 8066, 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, 4033, F₈, 2, 0) (dual of [(4033, 2), 8066, 1]-NRT-code) (see above)
    3. linear OOA(8⁰, 4033, F₈, 2, 0) (dual of [(4033, 2), 8066, 1]-NRT-code) (see above)
    4. 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]
    5. linear OOA(8¹, 4033, F₈, 2, 1) (dual of [(4033, 2), 8065, 2]-NRT-code) (see above)
    6. linear OOA(8¹, 4033, F₈, 2, 1) (dual of [(4033, 2), 8065, 2]-NRT-code) (see above)
    7. 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]
    8. 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]

(22−5, 22, 40833)-Net over F₈ — Digital

Digital (17, 22, 40833)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8²², 40833, F₈, 5) (dual of [40833, 40811, 6]-code), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OA(8⁰, 4681, F₈, 0) (dual of [4681, 4681, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]
  2. linear OA(8⁰, 4681, F₈, 0) (dual of [4681, 4681, 1]-code) (see above)
  3. linear OA(8⁰, 4681, F₈, 0) (dual of [4681, 4681, 1]-code) (see above)
  4. 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]
  5. linear OA(8¹, 4681, F₈, 1) (dual of [4681, 4680, 2]-code) (see above)
  6. linear OA(8¹, 4681, F₈, 1) (dual of [4681, 4680, 2]-code) (see above)
  7. linear OA(8⁵, 4681, F₈, 2) (dual of [4681, 4676, 3]-code), using
    - Hamming code H(5,8) [i]
  8. 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]

(22−5, 22, 65353)-Net in Base 8 — Constructive

(17, 22, 65353)-net in base 8, using

net defined by OOA [i] based on OOA(8²², 65353, S₈, 6, 5), using
- OOA stacking with additional row [i] based on OOA(8²², 65354, S₈, 2, 5), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(8³, 73, F₈, 2, 2) (dual of [(73, 2), 143, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(8³, 73, F₈, 2) (dual of [73, 70, 3]-code), using
        Hamming code H(3,8) [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]

(22−5, 22, 127024)-Net in Base 8

(17, 22, 127024)-net in base 8, using

net defined by OOA [i] based on OOA(8²², 127024, S₈, 6, 5), using
- OOA stacking with additional row [i] based on OOA(8²², 127025, S₈, 2, 5), using
  - discarding parts of the base [i] based on linear OOA(16¹⁶, 127025, F₁₆, 2, 5) (dual of [(127025, 2), 254034, 6]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁶, 127025, F₁₆, 5) (dual of [127025, 127009, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(16¹⁶, 130579, F₁₆, 5) (dual of [130579, 130563, 6]-code), using
        (u,Â u+v)-construction [i] based on
        linear OA(16², 17, F₁₆, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
        extended Reedâ€“Solomon code RS_e(15,16) [i]
        Hamming code H(2,16) [i]
        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]

(22−5, 22, large)-Net in Base 8 — Upper bound on s

There is no (17, 22, large)-net in base 8, because

3 times m-reduction [i] would yield (17, 19, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]