MinT - Best Known (114−21, 114, s)-Nets in Base 16

Best Known (114−21, 114, s)-Nets in Base 16

(114−21, 114, 104924)-Net over F₁₆ — Constructive and digital

Digital (93, 114, 104924)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 16, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (77, 98, 104859)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹⁸, 104859, F₁₆, 21, 21) (dual of [(104859, 21), 2201941, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(16⁹⁸, 1048591, F₁₆, 21) (dual of [1048591, 1048493, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁹⁸, 1048593, F₁₆, 21) (dual of [1048593, 1048495, 22]-code), using
        constructionÂ X applied to C_e(20) âŠ‚ C_e(17) [i] based on
        linear OA(16⁹⁶, 1048576, F₁₆, 21) (dual of [1048576, 1048480, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(16⁸¹, 1048576, F₁₆, 18) (dual of [1048576, 1048495, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(16², 17, F₁₆, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
        extended Reedâ€“Solomon code RS_e(15,16) [i]
        Hamming code H(2,16) [i]

(114−21, 114, 209717)-Net in Base 16 — Constructive

(93, 114, 209717)-net in base 16, using

net defined by OOA [i] based on OOA(16¹¹⁴, 209717, S₁₆, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(16¹¹⁴, 2097171, S₁₆, 21), using
  - discarding parts of the base [i] based on linear OA(128⁶⁵, 2097171, F₁₂₈, 21) (dual of [2097171, 2097106, 22]-code), using
    - constructionÂ X applied to C_e(20) âŠ‚ C_e(15) [i] based on
      1. linear OA(128⁶¹, 2097152, F₁₂₈, 21) (dual of [2097152, 2097091, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      2. linear OA(128⁴⁶, 2097152, F₁₂₈, 16) (dual of [2097152, 2097106, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      3. linear OA(128⁴, 19, F₁₂₈, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁴, 128, F₁₂₈, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
        Reedâ€“Solomon code RS(124,128) [i]

(114−21, 114, 4042965)-Net over F₁₆ — Digital

Digital (93, 114, 4042965)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(114−21, 114, large)-Net in Base 16 — Upper bound on s

There is no (93, 114, large)-net in base 16, because

19 times m-reduction [i] would yield (93, 95, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (114−21, 114, s)-Nets in Base 16

(114−21, 114, 104924)-Net over F16 — Constructive and digital

(114−21, 114, 209717)-Net in Base 16 — Constructive

(114−21, 114, 4042965)-Net over F16 — Digital

(114−21, 114, large)-Net in Base 16 — Upper bound on s

(114−21, 114, 104924)-Net over F₁₆ — Constructive and digital

(114−21, 114, 4042965)-Net over F₁₆ — Digital