MinT - Best Known (37−13, 37, s)-Nets in Base 128

Best Known (37−13, 37, s)-Nets in Base 128

(37−13, 37, 349525)-Net over F₁₂₈ — Constructive and digital

Digital (24, 37, 349525)-net over F₁₂₈, using

net defined by OOA [i] based on linear OOA(128³⁷, 349525, F₁₂₈, 13, 13) (dual of [(349525, 13), 4543788, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(128³⁷, 2097151, F₁₂₈, 13) (dual of [2097151, 2097114, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(128³⁷, 2097152, F₁₂₈, 13) (dual of [2097152, 2097115, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(37−13, 37, 846068)-Net over F₁₂₈ — Digital

Digital (24, 37, 846068)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128³⁷, 846068, F₁₂₈, 2, 13) (dual of [(846068, 2), 1692099, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(128³⁷, 1048577, F₁₂₈, 2, 13) (dual of [(1048577, 2), 2097117, 14]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(128³⁷, 2097154, F₁₂₈, 13) (dual of [2097154, 2097117, 14]-code), using
    - discarding factors / shortening the dual code based on linear OA(128³⁷, 2097155, F₁₂₈, 13) (dual of [2097155, 2097118, 14]-code), using
      - constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(128³⁷, 2097152, F₁₂₈, 13) (dual of [2097152, 2097115, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(128³⁴, 2097152, F₁₂₈, 12) (dual of [2097152, 2097118, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(37−13, 37, large)-Net in Base 128 — Upper bound on s

There is no (24, 37, large)-net in base 128, because

11 times m-reduction [i] would yield (24, 26, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]

Best Known (37−13, 37, s)-Nets in Base 128

(37−13, 37, 349525)-Net over F128 — Constructive and digital

(37−13, 37, 846068)-Net over F128 — Digital

(37−13, 37, large)-Net in Base 128 — Upper bound on s

(37−13, 37, 349525)-Net over F₁₂₈ — Constructive and digital

(37−13, 37, 846068)-Net over F₁₂₈ — Digital