MinT - Best Known (77, 77+23, s)-Nets in Base 4

Best Known (77, 77+23, s)-Nets in Base 4

Digital (77, 100, 1036)-net over F₄, using

trace code for nets [i] based on digital (2, 25, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

Digital (77, 100, 1657)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁰, 1657, F₄, 23) (dual of [1657, 1557, 24]-code), using
- 1556 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 33 times 0, 1, 34 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 45 times 0, 1, 49 times 0, 1, 51 times 0, 1, 55 times 0, 1, 59 times 0, 1, 63 times 0, 1, 67 times 0, 1, 71 times 0, 1, 76 times 0, 1, 81 times 0, 1, 87 times 0, 1, 92 times 0, 1, 99 times 0) [i] based on linear OA(4²³, 24, F₄, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
  - dual of repetition code with length 24 [i]

There is no (77, 100, 428968)-net in base 4, because

1 times m-reduction [i] would yield (77, 99, 428968)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 401743 127152 534906 391516 847124 395672 746106 360351 225052 320580 > 4⁹⁹ [i]