MinT - Best Known (171, 171+48, s)-Nets in Base 4

Best Known (171, 171+48, s)-Nets in Base 4

(171, 171+48, 1052)-Net over F₄ — Constructive and digital

Digital (171, 219, 1052)-net over F₄, using

1 times m-reduction [i] based on digital (171, 220, 1052)-net over F₄, using
- trace code for nets [i] based on digital (6, 55, 263)-net over F₂₅₆, using
  - net from sequence [i] based on digital (6, 262)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(171, 171+48, 4117)-Net over F₄ — Digital

Digital (171, 219, 4117)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁹, 4117, F₄, 48) (dual of [4117, 3898, 49]-code), using
- 7 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 6 times 0) [i] based on linear OA(4²¹⁸, 4109, F₄, 48) (dual of [4109, 3891, 49]-code), using
  - constructionÂ X applied to C_e(48) âŠ‚ C_e(45) [i] based on
    1. 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]
    2. linear OA(4²⁰⁵, 4096, F₄, 46) (dual of [4096, 3891, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [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]

(171, 171+48, 1018663)-Net in Base 4 — Upper bound on s

There is no (171, 219, 1018664)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 709817 928411 104576 531514 701379 493450 651535 508946 390330 632693 324782 401354 473192 953113 240746 426551 047967 292791 294092 533907 684355 651764 > 4²¹⁹ [i]