MinT - Best Known (158, 169, s)-Nets in Base 2

Best Known (158, 169, s)-Nets in Base 2

(158, 169, 4202494)-Net over F₂ — Constructive and digital

Digital (158, 169, 4202494)-net over F₂, using

2¹ times duplication [i] based on digital (157, 168, 4202494)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (23, 28, 8193)-net over F₂, using
    - nets constructed from net-embeddable BCH codes [i]
  2. digital (129, 140, 4194301)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2¹⁴⁰, 4194301, F₂, 15, 11) (dual of [(4194301, 15), 62914375, 12]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OOA(2¹⁴⁰, large, F₂, 3, 11), using
        OOAs constructed from BCH codes [i]

(158, 169, 4202500)-Net in Base 2 — Constructive

(158, 169, 4202500)-net in base 2, using

(u,Â u+v)-construction [i] based on
1. (24, 29, 8199)-net in base 2, using
  - net defined by OOA [i] based on OOA(2²⁹, 8199, S₂, 5, 5), using
    - OOA 2-folding and stacking with additional row [i] based on OA(2²⁹, 16399, S₂, 5), using
      - constructionÂ X4 applied to RM(1,14) âŠ‚ K(14) [i] based on
        OA(2²⁸, 16384, S₂, 5), using Kerdock OA K(14) [i]
        linear OA(2¹⁵, 16384, F₂, 3) (dual of [16384, 16369, 4]-code or 16384-cap in PG(14,2)), using Reedâ€“Muller code RM(1,14) [i]
        linear OA(2¹⁴, 15, F₂, 13) (dual of [15, 1, 14]-code), using
        strength reduction [i] based on linear OA(2¹⁴, 15, F₂, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
        dual of repetition code with length 15 [i]
        linear OA(2¹, 15, F₂, 1) (dual of [15, 14, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. digital (129, 140, 4194301)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁴⁰, 4194301, F₂, 15, 11) (dual of [(4194301, 15), 62914375, 12]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(2¹⁴⁰, large, F₂, 3, 11), using
      - OOAs constructed from BCH codes [i]

(158, 169, large)-Net over F₂ — Digital

Digital (158, 169, large)-net over F₂, using

2¹⁷ times duplication [i] based on digital (141, 152, large)-net over F₂, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁵², large, F₂, 3, 11), using
  - 4 times NRT-code embedding in larger space [i] based on linear OOA(2¹⁴⁰, large, F₂, 3, 11), using
    - OOAs constructed from BCH codes [i]

(158, 169, large)-Net in Base 2 — Upper bound on s

There is no (158, 169, large)-net in base 2, because

9 times m-reduction [i] would yield (158, 160, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]