MinT - Best Known (182, 232, s)-Nets in Base 4

Best Known (182, 232, s)-Nets in Base 4

(182, 232, 1060)-Net over F₄ — Constructive and digital

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]

(182, 232, 4544)-Net over F₄ — Digital

Digital (182, 232, 4544)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³², 4544, F₄, 50) (dual of [4544, 4312, 51]-code), using
- 433 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) [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]

(182, 232, 1311035)-Net in Base 4 — Upper bound on s

There is no (182, 232, 1311036)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 634269 390183 114768 653520 248923 504582 027022 097883 553592 116433 160335 649073 075961 613751 930029 945157 861561 783142 580666 413633 935781 427370 432987 > 4²³² [i]