MinT - Best Known (233−14, 233, s)-Nets in Base 4

Best Known (233−14, 233, s)-Nets in Base 4

(233−14, 233, 7624250)-Net over F₄ — Constructive and digital

Digital (219, 233, 7624250)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (66, 73, 2830766)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁷³, 2830766, F₄, 9, 7) (dual of [(2830766, 9), 25476821, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(4⁷³, 2830767, F₄, 3, 7) (dual of [(2830767, 3), 8492228, 8]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(4¹², 34566, F₄, 3, 3) (dual of [(34566, 3), 103686, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(4¹², 34566, F₄, 2, 3) (dual of [(34566, 2), 69120, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4¹², 34566, F₄, 3) (dual of [34566, 34554, 4]-code or 34566-cap in PG(11,4)), using
        product of projective cap with cap in AG(5,Â 4) [i] based on linear OA(4⁷, 288, F₄, 3) (dual of [288, 281, 4]-code or 288-cap in PG(6,4)), using
        product of two caps with tangent hyperplane [i]
        linear OOA(4⁶¹, 2796201, F₄, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(4⁶¹, large, F₄, 7) (dual of [large, large−61, 8]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [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]

(233−14, 233, large)-Net over F₄ — Digital

Digital (219, 233, large)-net over F₄, using

t-expansion [i] based on digital (213, 233, large)-net over F₄, using
- 4 times m-reduction [i] based on digital (213, 237, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³⁷, large, F₄, 24) (dual of [large, large−237, 25]-code), using
    - 21 times code embedding in larger space [i] based on linear OA(4²¹⁶, large, F₄, 24) (dual of [large, large−216, 25]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(233−14, 233, large)-Net in Base 4 — Upper bound on s

There is no (219, 233, large)-net in base 4, because

12 times m-reduction [i] would yield (219, 221, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (233−14, 233, s)-Nets in Base 4

(233−14, 233, 7624250)-Net over F4 — Constructive and digital

(233−14, 233, large)-Net over F4 — Digital

(233−14, 233, large)-Net in Base 4 — Upper bound on s

(233−14, 233, 7624250)-Net over F₄ — Constructive and digital

(233−14, 233, large)-Net over F₄ — Digital