MinT - Best Known (180, 231, s)-Nets in Base 4

Best Known (180, 231, s)-Nets in Base 4

(180, 231, 1052)-Net over F₄ — Constructive and digital

Digital (180, 231, 1052)-net over F₄, using

1 times m-reduction [i] based on digital (180, 232, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 58, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(180, 231, 4119)-Net over F₄ — Digital

Digital (180, 231, 4119)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³¹, 4119, F₄, 51) (dual of [4119, 3888, 52]-code), using
- 8 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 7 times 0) [i] based on linear OA(4²³⁰, 4110, F₄, 51) (dual of [4110, 3880, 52]-code), using
  - constructionÂ X applied to C([0,25]) âŠ‚ C([0,24]) [i] based on
    1. linear OA(4²²⁹, 4097, F₄, 51) (dual of [4097, 3868, 52]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,25], and minimum distance dÂ â‰¥ |{−25,−24,…,25}|+1Â =Â 52 (BCH-bound) [i]
    2. linear OA(4²¹⁷, 4097, F₄, 49) (dual of [4097, 3880, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,24], and minimum distance dÂ â‰¥ |{−24,−23,…,24}|+1Â =Â 50 (BCH-bound) [i]
    3. linear OA(4¹, 13, F₄, 1) (dual of [13, 12, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(180, 231, 1173407)-Net in Base 4 — Upper bound on s

There is no (180, 231, 1173408)-net in base 4, because

1 times m-reduction [i] would yield (180, 230, 1173408)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 977144 727574 110578 857759 426286 910026 909359 437078 117141 067806 970871 384862 696228 412589 693236 327445 020216 662077 684961 278656 230749 384647 070927 > 4²³⁰ [i]