MinT - Best Known (98−10, 98, s)-Nets in Base 9

Best Known (98−10, 98, s)-Nets in Base 9

(98−10, 98, 3413858)-Net over F₉ — Constructive and digital

Digital (88, 98, 3413858)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (19, 24, 58418)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9²⁴, 58418, F₉, 6, 5) (dual of [(58418, 6), 350484, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(9²⁴, 58419, F₉, 2, 5) (dual of [(58419, 2), 116814, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(9⁰, 6491, F₉, 2, 0) (dual of [(6491, 2), 12982, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9⁰, s, F₉, 2, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(9⁰, 6491, F₉, 2, 0) (dual of [(6491, 2), 12982, 1]-NRT-code) (see above)
        linear OOA(9⁰, 6491, F₉, 2, 0) (dual of [(6491, 2), 12982, 1]-NRT-code) (see above)
        linear OOA(9⁰, 6491, F₉, 2, 0) (dual of [(6491, 2), 12982, 1]-NRT-code) (see above)
        linear OOA(9¹, 6491, F₉, 2, 1) (dual of [(6491, 2), 12981, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9¹, s, F₉, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(9¹, 6491, F₉, 2, 1) (dual of [(6491, 2), 12981, 2]-NRT-code) (see above)
        linear OOA(9¹, 6491, F₉, 2, 1) (dual of [(6491, 2), 12981, 2]-NRT-code) (see above)
        linear OOA(9⁵, 6491, F₉, 2, 2) (dual of [(6491, 2), 12977, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9⁵, 7381, F₉, 2, 2) (dual of [(7381, 2), 14757, 3]-NRT-code), using
        appending kth column [i] based on linear OA(9⁵, 7381, F₉, 2) (dual of [7381, 7376, 3]-code), using
        Hamming code H(5,9) [i]
        linear OOA(9¹⁶, 6491, F₉, 2, 5) (dual of [(6491, 2), 12966, 6]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(9², 10, F₉, 2, 2) (dual of [(10, 2), 18, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;18,9) [i]
        linear OOA(9¹⁴, 6481, F₉, 2, 5) (dual of [(6481, 2), 12948, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(9¹⁴, 12962, F₉, 5) (dual of [12962, 12948, 6]-code), using
        trace code [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (64, 74, 3355440)-net over F₉, using
  - trace code for nets [i] based on digital (27, 37, 1677720)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81³⁷, 1677720, F₈₁, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(81³⁷, 8388600, F₈₁, 10) (dual of [8388600, 8388563, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(81³⁷, large, F₈₁, 10) (dual of [large, large−37, 11]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(98−10, 98, large)-Net over F₉ — Digital

Digital (88, 98, large)-net over F₉, using

9¹ times duplication [i] based on digital (87, 97, large)-net over F₉, using
- t-expansion [i] based on digital (83, 97, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(98−10, 98, large)-Net in Base 9 — Upper bound on s

There is no (88, 98, large)-net in base 9, because

8 times m-reduction [i] would yield (88, 90, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (98−10, 98, s)-Nets in Base 9

(98−10, 98, 3413858)-Net over F9 — Constructive and digital

(98−10, 98, large)-Net over F9 — Digital

(98−10, 98, large)-Net in Base 9 — Upper bound on s

(98−10, 98, 3413858)-Net over F₉ — Constructive and digital

(98−10, 98, large)-Net over F₉ — Digital