MinT - Best Known (150, 150+12, s)-Nets in Base 4

Best Known (150, 150+12, s)-Nets in Base 4

(150, 150+12, 5593768)-Net over F₄ — Constructive and digital

Digital (150, 162, 5593768)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (20, 26, 1368)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²⁶, 1368, F₄, 6, 6) (dual of [(1368, 6), 8182, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(4²⁶, 4104, F₄, 6) (dual of [4104, 4078, 7]-code), using
      - constructionÂ X4 applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(4²⁵, 4096, F₄, 6) (dual of [4096, 4071, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4¹⁹, 4096, F₄, 5) (dual of [4096, 4077, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(4⁷, 8, F₄, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
        dual of repetition code with length 8 [i]
        linear OA(4¹, 8, F₄, 1) (dual of [8, 7, 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]
2. digital (124, 136, 5592400)-net over F₄, using
  - trace code for nets [i] based on digital (56, 68, 2796200)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁶⁸, 2796200, F₁₆, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶⁸, 8388601, F₁₆, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁶⁸, 8388602, F₁₆, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
        trace code [i] based on linear OOA(256³⁴, 4194301, F₂₅₆, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁴, 8388602, F₂₅₆, 12) (dual of [8388602, 8388568, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(150, 150+12, large)-Net over F₄ — Digital

Digital (150, 162, large)-net over F₄, using

t-expansion [i] based on digital (149, 162, large)-net over F₄, using
- 4 times m-reduction [i] based on digital (149, 166, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁶⁶, large, F₄, 17) (dual of [large, large−166, 18]-code), using
    - 21 times code embedding in larger space [i] based on linear OA(4¹⁴⁵, large, F₄, 17) (dual of [large, large−145, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(150, 150+12, large)-Net in Base 4 — Upper bound on s

There is no (150, 162, large)-net in base 4, because

10 times m-reduction [i] would yield (150, 152, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (150, 150+12, s)-Nets in Base 4

(150, 150+12, 5593768)-Net over F4 — Constructive and digital

(150, 150+12, large)-Net over F4 — Digital

(150, 150+12, large)-Net in Base 4 — Upper bound on s

(150, 150+12, 5593768)-Net over F₄ — Constructive and digital

(150, 150+12, large)-Net over F₄ — Digital