MinT - Best Known (81, 91, s)-Nets in Base 7

Best Known (81, 91, s)-Nets in Base 7

(81, 91, 2308331)-Net over F₇ — Constructive and digital

Digital (81, 91, 2308331)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (12, 17, 2409)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7¹⁷, 2409, F₇, 6, 5) (dual of [(2409, 6), 14437, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(7¹⁷, 2410, F₇, 2, 5) (dual of [(2410, 2), 4803, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(7³, 57, F₇, 2, 2) (dual of [(57, 2), 111, 3]-NRT-code), using
        appending kth column [i] based on linear OA(7³, 57, F₇, 2) (dual of [57, 54, 3]-code), using
        Hamming code H(3,7) [i]
        linear OOA(7¹⁴, 2353, F₇, 2, 5) (dual of [(2353, 2), 4692, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(7¹⁴, 4706, F₇, 5) (dual of [4706, 4692, 6]-code), using
        trace code [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (64, 74, 2305922)-net over F₇, using
  - trace code for nets [i] based on digital (27, 37, 1152961)-net over F₄₉, using
    - net defined by OOA [i] based on linear OOA(49³⁷, 1152961, F₄₉, 10, 10) (dual of [(1152961, 10), 11529573, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(49³⁷, 5764805, F₄₉, 10) (dual of [5764805, 5764768, 11]-code), using
        constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
        linear OA(49³⁷, 5764801, F₄₉, 10) (dual of [5764801, 5764764, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(49³³, 5764801, F₄₉, 9) (dual of [5764801, 5764768, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(81, 91, large)-Net over F₇ — Digital

Digital (81, 91, large)-net over F₇, using

7¹ times duplication [i] based on digital (80, 90, large)-net over F₇, using
- t-expansion [i] based on digital (78, 90, large)-net over F₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁹⁰, large, F₇, 12) (dual of [large, large−90, 13]-code), using
    - trace code [i] based on linear OA(49⁴⁵, 5764801, F₄₉, 12) (dual of [5764801, 5764756, 13]-code), using
      - an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(81, 91, large)-Net in Base 7 — Upper bound on s

There is no (81, 91, large)-net in base 7, because

8 times m-reduction [i] would yield (81, 83, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (81, 91, s)-Nets in Base 7

(81, 91, 2308331)-Net over F7 — Constructive and digital

(81, 91, large)-Net over F7 — Digital

(81, 91, large)-Net in Base 7 — Upper bound on s

(81, 91, 2308331)-Net over F₇ — Constructive and digital

(81, 91, large)-Net over F₇ — Digital