MinT - Best Known (42, 42+32, s)-Nets in Base 16

Best Known (42, 42+32, s)-Nets in Base 16

(42, 42+32, 524)-Net over F₁₆ — Constructive and digital

Digital (42, 74, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 37, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(42, 42+32, 669)-Net over F₁₆ — Digital

Digital (42, 74, 669)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷⁴, 669, F₁₆, 32) (dual of [669, 595, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁷⁴, 673, F₁₆, 32) (dual of [673, 599, 33]-code), using
  - 25 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 5 times 0, 1, 15 times 0) [i] based on linear OA(16⁶⁸, 642, F₁₆, 32) (dual of [642, 574, 33]-code), using
    - trace code [i] based on linear OA(256³⁴, 321, F₂₅₆, 32) (dual of [321, 287, 33]-code), using
      - extended algebraic-geometric code AG_e(F,288P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
        sharp bound on the number of rational points on curves with genus 2 [i]

(42, 42+32, 168066)-Net in Base 16 — Upper bound on s

There is no (42, 74, 168067)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 127325 060979 501746 136934 081601 173025 708968 465424 191419 532320 664370 394363 934645 904847 286456 > 16⁷⁴ [i]