MinT - Best Known (64−15, 64, s)-Nets in Base 128

Best Known (64−15, 64, s)-Nets in Base 128

(64−15, 64, 1198500)-Net over F₁₂₈ — Constructive and digital

Digital (49, 64, 1198500)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (42, 57, 1198371)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁵⁷, 1198371, F₁₂₈, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(128⁵⁷, 8388598, F₁₂₈, 15) (dual of [8388598, 8388541, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁵⁷, large, F₁₂₈, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 15790321Â | 128⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(64−15, 64, 1220216)-Net in Base 128 — Constructive

(49, 64, 1220216)-net in base 128, using

base change [i] based on digital (41, 56, 1220216)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 13, 21845)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256¹³, 21845, F₂₅₆, 7, 7) (dual of [(21845, 7), 152902, 8]-NRT-code), using
      - appending kth column [i] based on linear OOA(256¹³, 21845, F₂₅₆, 6, 7) (dual of [(21845, 6), 131057, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(256¹³, 65536, F₂₅₆, 7) (dual of [65536, 65523, 8]-code), using
        an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  2. 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]

(64−15, 64, large)-Net over F₁₂₈ — Digital

Digital (49, 64, large)-net over F₁₂₈, using

t-expansion [i] based on digital (48, 64, large)-net over F₁₂₈, using
- 2 times m-reduction [i] based on digital (48, 66, large)-net over F₁₂₈, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(64−15, 64, large)-Net in Base 128 — Upper bound on s

There is no (49, 64, large)-net in base 128, because

13 times m-reduction [i] would yield (49, 51, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (64−15, 64, s)-Nets in Base 128

(64−15, 64, 1198500)-Net over F128 — Constructive and digital

(64−15, 64, 1220216)-Net in Base 128 — Constructive

(64−15, 64, large)-Net over F128 — Digital

(64−15, 64, large)-Net in Base 128 — Upper bound on s

(64−15, 64, 1198500)-Net over F₁₂₈ — Constructive and digital

(64−15, 64, large)-Net over F₁₂₈ — Digital