MinT - Best Known (63, 77, s)-Nets in Base 81

Best Known (63, 77, s)-Nets in Base 81

(63, 77, 1391836)-Net over F₈₁ — Constructive and digital

Digital (63, 77, 1391836)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (17, 24, 193465)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81²⁴, 193465, F₈₁, 9, 7) (dual of [(193465, 9), 1741161, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(81²⁴, 193466, F₈₁, 3, 7) (dual of [(193466, 3), 580374, 8]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(81⁵, 16318, F₈₁, 3, 3) (dual of [(16318, 3), 48949, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(81⁵, 16318, F₈₁, 2, 3) (dual of [(16318, 2), 32631, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁵, 16318, F₈₁, 3) (dual of [16318, 16313, 4]-code or 16318-cap in PG(4,81)), using
        large caps in PG(4,Â b) [i]
        linear OOA(81¹⁹, 177148, F₈₁, 3, 7) (dual of [(177148, 3), 531425, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(81¹⁹, 531444, F₈₁, 7) (dual of [531444, 531425, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(81¹⁹, 531441, F₈₁, 7) (dual of [531441, 531422, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(81¹⁶, 531441, F₈₁, 6) (dual of [531441, 531425, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (39, 53, 1198371)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁵³, 1198371, F₈₁, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(81⁵³, 8388597, F₈₁, 14) (dual of [8388597, 8388544, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁵³, large, F₈₁, 14) (dual of [large, large−53, 15]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(63, 77, large)-Net over F₈₁ — Digital

Digital (63, 77, large)-net over F₈₁, using

t-expansion [i] based on digital (60, 77, large)-net over F₈₁, using
- 4 times m-reduction [i] based on digital (60, 81, large)-net over F₈₁, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81⁸¹, large, F₈₁, 21) (dual of [large, large−81, 22]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(63, 77, large)-Net in Base 81 — Upper bound on s

There is no (63, 77, large)-net in base 81, because

12 times m-reduction [i] would yield (63, 65, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (63, 77, s)-Nets in Base 81

(63, 77, 1391836)-Net over F81 — Constructive and digital

(63, 77, large)-Net over F81 — Digital

(63, 77, large)-Net in Base 81 — Upper bound on s

(63, 77, 1391836)-Net over F₈₁ — Constructive and digital

(63, 77, large)-Net over F₈₁ — Digital