MinT - Best Known (169−35, 169, s)-Nets in Base 4

Best Known (169−35, 169, s)-Nets in Base 4

(169−35, 169, 1056)-Net over F₄ — Constructive and digital

Digital (134, 169, 1056)-net over F₄, using

4¹ times duplication [i] based on digital (133, 168, 1056)-net over F₄, using
- trace code for nets [i] based on digital (7, 42, 264)-net over F₂₅₆, using
  - net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(169−35, 169, 4512)-Net over F₄ — Digital

Digital (134, 169, 4512)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁶⁹, 4512, F₄, 35) (dual of [4512, 4343, 36]-code), using
- 398 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 28 times 0, 1, 44 times 0, 1, 66 times 0, 1, 94 times 0, 1, 123 times 0) [i] based on linear OA(4¹⁵⁷, 4102, F₄, 35) (dual of [4102, 3945, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(33) [i] based on
    1. linear OA(4¹⁵⁷, 4096, F₄, 35) (dual of [4096, 3939, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    2. linear OA(4¹⁵¹, 4096, F₄, 34) (dual of [4096, 3945, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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]

(169−35, 169, 2131034)-Net in Base 4 — Upper bound on s

There is no (134, 169, 2131035)-net in base 4, because

1 times m-reduction [i] would yield (134, 168, 2131035)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 139984 537714 493677 919803 453697 898358 758538 657031 974344 793119 838198 940294 935249 544046 216536 851576 293238 > 4¹⁶⁸ [i]