MinT - Best Known (245−15, 245, s)-Nets in Base 4

Best Known (245−15, 245, s)-Nets in Base 4

(245−15, 245, 7624250)-Net over F₄ — Constructive and digital

Digital (230, 245, 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 (157, 172, 4793484)-net over F₄, using
  - trace code for nets [i] based on digital (28, 43, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴³, 1198371, F₂₅₆, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(256⁴³, 8388598, F₂₅₆, 15) (dual of [8388598, 8388555, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴³, large, F₂₅₆, 15) (dual of [large, large−43, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(245−15, 245, large)-Net over F₄ — Digital

Digital (230, 245, large)-net over F₄, using

t-expansion [i] based on digital (222, 245, large)-net over F₄, using
- 2 times m-reduction [i] based on digital (222, 247, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴⁷, large, F₄, 25) (dual of [large, large−247, 26]-code), using
    - 30 times code embedding in larger space [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]

(245−15, 245, large)-Net in Base 4 — Upper bound on s

There is no (230, 245, large)-net in base 4, because

13 times m-reduction [i] would yield (230, 232, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (245−15, 245, s)-Nets in Base 4

(245−15, 245, 7624250)-Net over F4 — Constructive and digital

(245−15, 245, large)-Net over F4 — Digital

(245−15, 245, large)-Net in Base 4 — Upper bound on s

(245−15, 245, 7624250)-Net over F₄ — Constructive and digital

(245−15, 245, large)-Net over F₄ — Digital