MinT - Best Known (183, 183+50, s)-Nets in Base 4

Best Known (183, 183+50, s)-Nets in Base 4

(183, 183+50, 1060)-Net over F₄ — Constructive and digital

Digital (183, 233, 1060)-net over F₄, using

4¹ times duplication [i] based on digital (182, 232, 1060)-net over F₄, using
- trace code for nets [i] based on digital (8, 58, 265)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 264)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(183, 183+50, 4671)-Net over F₄ — Digital

Digital (183, 233, 4671)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³³, 4671, F₄, 50) (dual of [4671, 4438, 51]-code), using
- 559 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 7 times 0, 1, 18 times 0, 1, 35 times 0, 1, 58 times 0, 1, 83 times 0, 1, 104 times 0, 1, 118 times 0, 1, 125 times 0) [i] based on linear OA(4²²³, 4102, F₄, 50) (dual of [4102, 3879, 51]-code), using
  - constructionÂ X applied to C_e(49) âŠ‚ C_e(48) [i] based on
    1. 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]
    2. linear OA(4²¹⁷, 4096, F₄, 49) (dual of [4096, 3879, 50]-code), using an extension C_e(48) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,48], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 49 [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]

(183, 183+50, 1385789)-Net in Base 4 — Upper bound on s

There is no (183, 233, 1385790)-net in base 4, because

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]