MinT - Best Known (191, 191+14, s)-Nets in Base 4

Best Known (191, 191+14, s)-Nets in Base 4

(191, 191+14, 4815348)-Net over F₄ — Constructive and digital

Digital (191, 205, 4815348)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (38, 45, 21864)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁴⁵, 21864, F₄, 9, 7) (dual of [(21864, 9), 196731, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(4⁴⁵, 21865, F₄, 3, 7) (dual of [(21865, 3), 65550, 8]-NRT-code), using
      - (u,Â u+v)-construction [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]
        linear OOA(4⁴¹, 21848, F₄, 3, 7) (dual of [(21848, 3), 65503, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(4⁴¹, 65544, F₄, 7) (dual of [65544, 65503, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(4⁴¹, 65536, F₄, 7) (dual of [65536, 65495, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(4³³, 65536, F₄, 6) (dual of [65536, 65503, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4⁰, 8, F₄, 0) (dual of [8, 8, 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]

(191, 191+14, large)-Net over F₄ — Digital

Digital (191, 205, large)-net over F₄, using

t-expansion [i] based on digital (186, 205, large)-net over F₄, using
- 2 times m-reduction [i] based on digital (186, 207, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁷, large, F₄, 21) (dual of [large, large−207, 22]-code), using
    - 26 times code embedding in larger space [i] based on linear OA(4¹⁸¹, large, F₄, 21) (dual of [large, large−181, 22]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(191, 191+14, large)-Net in Base 4 — Upper bound on s

There is no (191, 205, large)-net in base 4, because

12 times m-reduction [i] would yield (191, 193, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (191, 191+14, s)-Nets in Base 4

(191, 191+14, 4815348)-Net over F4 — Constructive and digital

(191, 191+14, large)-Net over F4 — Digital

(191, 191+14, large)-Net in Base 4 — Upper bound on s

(191, 191+14, 4815348)-Net over F₄ — Constructive and digital

(191, 191+14, large)-Net over F₄ — Digital