MinT - Best Known (20, 20+14, s)-Nets in Base 128

Best Known (20, 20+14, s)-Nets in Base 128

(20, 20+14, 2469)-Net over F₁₂₈ — Constructive and digital

Digital (20, 34, 2469)-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 (13, 27, 2340)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128²⁷, 2340, F₁₂₈, 14, 14) (dual of [(2340, 14), 32733, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(128²⁷, 16380, F₁₂₈, 14) (dual of [16380, 16353, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(128²⁷, 16384, F₁₂₈, 14) (dual of [16384, 16357, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(20, 20+14, 9363)-Net in Base 128 — Constructive

(20, 34, 9363)-net in base 128, using

1 times m-reduction [i] based on (20, 35, 9363)-net in base 128, using
- net defined by OOA [i] based on OOA(128³⁵, 9363, S₁₂₈, 15, 15), using
  - OOA 7-folding and stacking with additional row [i] based on OA(128³⁵, 65542, S₁₂₈, 15), using
    - discarding parts of the base [i] based on linear OA(256³⁰, 65542, F₂₅₆, 15) (dual of [65542, 65512, 16]-code), using
      - constructionÂ X applied to C([0,7]) âŠ‚ C([0,6]) [i] based on
        linear OA(256²⁹, 65537, F₂₅₆, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(256²⁵, 65537, F₂₅₆, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(256¹, 5, F₂₅₆, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(256¹, s, F₂₅₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(20, 20+14, 16515)-Net over F₁₂₈ — Digital

Digital (20, 34, 16515)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128³⁴, 16515, F₁₂₈, 14) (dual of [16515, 16481, 15]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(128⁷, 129, F₁₂₈, 7) (dual of [129, 122, 8]-code or 129-arc in PG(6,128)), using
    - extended Reedâ€“Solomon code RS_e(122,128) [i]
    - the expurgated narrow-sense BCH-code C(I) with length 129Â | 128²âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
  2. linear OA(128²⁷, 16386, F₁₂₈, 14) (dual of [16386, 16359, 15]-code), using
    - constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
      1. linear OA(128²⁷, 16384, F₁₂₈, 14) (dual of [16384, 16357, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
      2. linear OA(128²⁵, 16384, F₁₂₈, 13) (dual of [16384, 16359, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      3. linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 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]

(20, 20+14, 18618)-Net in Base 128

(20, 34, 18618)-net in base 128, using

128² times duplication [i] based on (18, 32, 18618)-net in base 128, using
- base change [i] based on digital (14, 28, 18618)-net over F₂₅₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256²⁸, 18618, F₂₅₆, 3, 14) (dual of [(18618, 3), 55826, 15]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(256²⁸, 21847, F₂₅₆, 3, 14) (dual of [(21847, 3), 65513, 15]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(256²⁸, 65541, F₂₅₆, 14) (dual of [65541, 65513, 15]-code), using
        constructionÂ X applied to C_e(13) âŠ‚ C_e(11) [i] based on
        linear OA(256²⁷, 65536, F₂₅₆, 14) (dual of [65536, 65509, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(256²³, 65536, F₂₅₆, 12) (dual of [65536, 65513, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(256¹, 5, F₂₅₆, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(256¹, s, F₂₅₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(20, 20+14, large)-Net in Base 128 — Upper bound on s

There is no (20, 34, large)-net in base 128, because

12 times m-reduction [i] would yield (20, 22, large)-net in base 128, but
- mutually orthogonal hypercube bound [i]