MinT - Best Known (42−8, 42, s)-Nets in Base 16

Best Known (42−8, 42, s)-Nets in Base 16

(42−8, 42, 262265)-Net over F₁₆ — Constructive and digital

Digital (34, 42, 262265)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 120)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁶, 120, F₁₆, 4, 4) (dual of [(120, 4), 474, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(16⁶, 240, F₁₆, 4) (dual of [240, 234, 5]-code), using
      - 1 times truncation [i] based on linear OA(16⁷, 241, F₁₆, 5) (dual of [241, 234, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (28, 36, 262145)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16³⁶, 262145, F₁₆, 8, 8) (dual of [(262145, 8), 2097124, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(16³⁶, 1048580, F₁₆, 8) (dual of [1048580, 1048544, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(16³⁶, 1048581, F₁₆, 8) (dual of [1048581, 1048545, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(16³⁶, 1048576, F₁₆, 8) (dual of [1048576, 1048540, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(16³¹, 1048576, F₁₆, 7) (dual of [1048576, 1048545, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(16⁰, 5, F₁₆, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(42−8, 42, 524290)-Net in Base 16 — Constructive

(34, 42, 524290)-net in base 16, using

base change [i] based on digital (16, 24, 524290)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128²⁴, 524290, F₁₂₈, 8, 8) (dual of [(524290, 8), 4194296, 9]-NRT-code), using
  - OA 4-folding and stacking [i] based on linear OA(128²⁴, 2097160, F₁₂₈, 8) (dual of [2097160, 2097136, 9]-code), using
    - discarding factors / shortening the dual code based on linear OA(128²⁴, 2097163, F₁₂₈, 8) (dual of [2097163, 2097139, 9]-code), using
      - constructionÂ X applied to C_e(7) âŠ‚ C_e(4) [i] based on
        linear OA(128²², 2097152, F₁₂₈, 8) (dual of [2097152, 2097130, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(128¹³, 2097152, F₁₂₈, 5) (dual of [2097152, 2097139, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(128², 11, F₁₂₈, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
        discarding factors / shortening the dual code based on linear OA(128², 128, F₁₂₈, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
        Reedâ€“Solomon code RS(126,128) [i]

(42−8, 42, 3780486)-Net over F₁₆ — Digital

Digital (34, 42, 3780486)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−8, 42, large)-Net in Base 16 — Upper bound on s

There is no (34, 42, large)-net in base 16, because

6 times m-reduction [i] would yield (34, 36, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (42−8, 42, s)-Nets in Base 16

(42−8, 42, 262265)-Net over F16 — Constructive and digital

(42−8, 42, 524290)-Net in Base 16 — Constructive

(42−8, 42, 3780486)-Net over F16 — Digital

(42−8, 42, large)-Net in Base 16 — Upper bound on s

(42−8, 42, 262265)-Net over F₁₆ — Constructive and digital

(42−8, 42, 3780486)-Net over F₁₆ — Digital