MinT - Best Known (80−15, 80, s)-Nets in Base 81

Best Known (80−15, 80, s)-Nets in Base 81

(80−15, 80, 1382080)-Net over F₈₁ — Constructive and digital

Digital (65, 80, 1382080)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (16, 23, 183709)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81²³, 183709, F₈₁, 9, 7) (dual of [(183709, 9), 1653358, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(81²³, 183710, F₈₁, 3, 7) (dual of [(183710, 3), 551107, 8]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(81⁴, 6562, F₈₁, 3, 3) (dual of [(6562, 3), 19682, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(81⁴, 6562, F₈₁, 2, 3) (dual of [(6562, 2), 13120, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁴, 6562, F₈₁, 3) (dual of [6562, 6558, 4]-code or 6562-cap in PG(3,81)), using
        ovoid in PG(3,Â 81) [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 (42, 57, 1198371)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁵⁷, 1198371, F₈₁, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(81⁵⁷, 8388598, F₈₁, 15) (dual of [8388598, 8388541, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(81⁵⁷, large, F₈₁, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(80−15, 80, large)-Net over F₈₁ — Digital

Digital (65, 80, large)-net over F₈₁, using

t-expansion [i] based on digital (60, 80, large)-net over F₈₁, using
- 1 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]

(80−15, 80, large)-Net in Base 81 — Upper bound on s

There is no (65, 80, large)-net in base 81, because

13 times m-reduction [i] would yield (65, 67, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]

Best Known (80−15, 80, s)-Nets in Base 81

(80−15, 80, 1382080)-Net over F81 — Constructive and digital

(80−15, 80, large)-Net over F81 — Digital

(80−15, 80, large)-Net in Base 81 — Upper bound on s

(80−15, 80, 1382080)-Net over F₈₁ — Constructive and digital

(80−15, 80, large)-Net over F₈₁ — Digital