MinT - Best Known (240−17, 240, s)-Nets in Base 4

Best Known (240−17, 240, s)-Nets in Base 4

(240−17, 240, 4198400)-Net over F₄ — Constructive and digital

Digital (223, 240, 4198400)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (36, 44, 4100)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁴⁴, 4100, F₄, 8, 8) (dual of [(4100, 8), 32756, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(4⁴⁴, 16400, F₄, 8) (dual of [16400, 16356, 9]-code), using
      - constructionÂ X4 applied to C_e(8) âŠ‚ C_e(5) [i] based on
        linear OA(4⁴³, 16384, F₄, 9) (dual of [16384, 16341, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(4²⁹, 16384, F₄, 6) (dual of [16384, 16355, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4¹⁵, 16, F₄, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,4)), using
        dual of repetition code with length 16 [i]
        linear OA(4¹, 16, F₄, 1) (dual of [16, 15, 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 (179, 196, 4194300)-net over F₄, using
  - trace code for nets [i] based on digital (81, 98, 2097150)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁹⁸, 2097150, F₁₆, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OOA(16⁹⁸, 8388601, F₁₆, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁹⁸, 8388602, F₁₆, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
        trace code [i] based on linear OOA(256⁴⁹, 4194301, F₂₅₆, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256⁴⁹, 8388602, F₂₅₆, 17) (dual of [8388602, 8388553, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁹, large, F₂₅₆, 17) (dual of [large, large−49, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(240−17, 240, large)-Net over F₄ — Digital

Digital (223, 240, large)-net over F₄, using

t-expansion [i] based on digital (222, 240, large)-net over F₄, using
- 7 times m-reduction [i] based on digital (222, 247, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴⁷, large, F₄, 25) (dual of [large, large−247, 26]-code), using
    - 30 times code embedding in larger space [i] based on linear OA(4²¹⁷, large, F₄, 25) (dual of [large, large−217, 26]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(240−17, 240, large)-Net in Base 4 — Upper bound on s

There is no (223, 240, large)-net in base 4, because

15 times m-reduction [i] would yield (223, 225, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (240−17, 240, s)-Nets in Base 4

(240−17, 240, 4198400)-Net over F4 — Constructive and digital

(240−17, 240, large)-Net over F4 — Digital

(240−17, 240, large)-Net in Base 4 — Upper bound on s

(240−17, 240, 4198400)-Net over F₄ — Constructive and digital

(240−17, 240, large)-Net over F₄ — Digital