MinT - Best Known (234−51, 234, s)-Nets in Base 4

Best Known (234−51, 234, s)-Nets in Base 4

(234−51, 234, 1056)-Net over F₄ — Constructive and digital

Digital (183, 234, 1056)-net over F₄, using

4² times duplication [i] based on digital (181, 232, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 58, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(234−51, 234, 4301)-Net over F₄ — Digital

Digital (183, 234, 4301)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³⁴, 4301, F₄, 51) (dual of [4301, 4067, 52]-code), using
- 194 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 1, 7 times 0, 1, 26 times 0, 1, 61 times 0, 1, 94 times 0) [i] based on linear OA(4²²⁹, 4102, F₄, 51) (dual of [4102, 3873, 52]-code), using
  - constructionÂ X applied to C_e(50) âŠ‚ C_e(49) [i] based on
    1. linear OA(4²²⁹, 4096, F₄, 51) (dual of [4096, 3867, 52]-code), using an extension C_e(50) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,50], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 51 [i]
    2. linear OA(4²²³, 4096, F₄, 50) (dual of [4096, 3873, 51]-code), using an extension C_e(49) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,49], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 50 [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]

(234−51, 234, 1385789)-Net in Base 4 — Upper bound on s

There is no (183, 234, 1385790)-net in base 4, because

1 times m-reduction [i] would yield (183, 233, 1385790)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 190 537547 855083 058308 520510 526614 576270 714005 061225 143149 356281 767671 628190 658464 567140 527487 433841 295635 176080 369196 937782 058295 254145 705307 > 4²³³ [i]