MinT - Best Known (215−14, 215, s)-Nets in Base 4

Best Known (215−14, 215, s)-Nets in Base 4

(215−14, 215, 5143029)-Net over F₄ — Constructive and digital

Digital (201, 215, 5143029)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (48, 55, 349545)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 17)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁴, 17, F₄, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(4⁴, 17, F₄, 2, 3) (dual of [(17, 2), 30, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [i]
    2. digital (44, 51, 349528)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁵¹, 349528, F₄, 7, 7) (dual of [(349528, 7), 2446645, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(4⁵¹, 1048585, F₄, 7) (dual of [1048585, 1048534, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁵¹, 1048586, F₄, 7) (dual of [1048586, 1048535, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(4⁵¹, 1048576, F₄, 7) (dual of [1048576, 1048525, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(4⁴¹, 1048576, F₄, 6) (dual of [1048576, 1048535, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4⁰, 10, F₄, 0) (dual of [10, 10, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 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 (146, 160, 4793484)-net over F₄, using
  - trace code for nets [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(215−14, 215, large)-Net over F₄ — Digital

Digital (201, 215, large)-net over F₄, using

t-expansion [i] based on digital (195, 215, large)-net over F₄, using
- 2 times m-reduction [i] based on digital (195, 217, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁷, large, F₄, 22) (dual of [large, large−217, 23]-code), using
    - strength reduction [i] based on linear OA(4²¹⁷, large, F₄, 25) (dual of [large, large−217, 26]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(215−14, 215, large)-Net in Base 4 — Upper bound on s

There is no (201, 215, large)-net in base 4, because

12 times m-reduction [i] would yield (201, 203, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (215−14, 215, s)-Nets in Base 4

(215−14, 215, 5143029)-Net over F4 — Constructive and digital

(215−14, 215, large)-Net over F4 — Digital

(215−14, 215, large)-Net in Base 4 — Upper bound on s

(215−14, 215, 5143029)-Net over F₄ — Constructive and digital

(215−14, 215, large)-Net over F₄ — Digital