MinT - Best Known (180, 180+50, s)-Nets in Base 4

Best Known (180, 180+50, s)-Nets in Base 4

(180, 180+50, 1056)-Net over F₄ — Constructive and digital

Digital (180, 230, 1056)-net over F₄, using

4² times duplication [i] based on digital (178, 228, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 57, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(180, 180+50, 4318)-Net over F₄ — Digital

Digital (180, 230, 4318)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³⁰, 4318, F₄, 50) (dual of [4318, 4088, 51]-code), using
- 209 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) [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]

(180, 180+50, 1173407)-Net in Base 4 — Upper bound on s

There is no (180, 230, 1173408)-net in base 4, because

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]