MinT - Best Known (148−31, 148, s)-Nets in Base 4

Best Known (148−31, 148, s)-Nets in Base 4

(148−31, 148, 1052)-Net over F₄ — Constructive and digital

Digital (117, 148, 1052)-net over F₄, using

trace code for nets [i] based on digital (6, 37, 263)-net over F₂₅₆, using
- net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(148−31, 148, 4200)-Net over F₄ — Digital

Digital (117, 148, 4200)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴⁸, 4200, F₄, 31) (dual of [4200, 4052, 32]-code), using
- 89 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 13 times 0, 1, 21 times 0, 1, 35 times 0) [i] based on linear OA(4¹³⁹, 4102, F₄, 31) (dual of [4102, 3963, 32]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(29) [i] based on
    1. linear OA(4¹³⁹, 4096, F₄, 31) (dual of [4096, 3957, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(4¹³³, 4096, F₄, 30) (dual of [4096, 3963, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    3. linear OA(4⁰, 6, F₄, 0) (dual of [6, 6, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(148−31, 148, 1701493)-Net in Base 4 — Upper bound on s

There is no (117, 148, 1701494)-net in base 4, because

1 times m-reduction [i] would yield (117, 147, 1701494)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 31828 889227 904353 677174 310613 588146 505314 855358 005610 851281 406290 388530 005952 340618 925752 > 4¹⁴⁷ [i]