MinT - Best Known (260−35, 260, s)-Nets in Base 4

Best Known (260−35, 260, s)-Nets in Base 4

(260−35, 260, 15442)-Net over F₄ — Constructive and digital

Digital (225, 260, 15442)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 24, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (201, 236, 15421)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²³⁶, 15421, F₄, 35, 35) (dual of [(15421, 35), 539499, 36]-NRT-code), using
    - OOA 17-folding and stacking with additional row [i] based on linear OA(4²³⁶, 262158, F₄, 35) (dual of [262158, 261922, 36]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²³⁶, 262164, F₄, 35) (dual of [262164, 261928, 36]-code), using
        constructionÂ X applied to C([0,17]) âŠ‚ C([0,16]) [i] based on
        linear OA(4²³⁵, 262145, F₄, 35) (dual of [262145, 261910, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
        linear OA(4²¹⁷, 262145, F₄, 33) (dual of [262145, 261928, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
        linear OA(4¹, 19, F₄, 1) (dual of [19, 18, 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]

(260−35, 260, 233046)-Net over F₄ — Digital

Digital (225, 260, 233046)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁶⁰, 233046, F₄, 35) (dual of [233046, 232786, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁶⁰, 262175, F₄, 35) (dual of [262175, 261915, 36]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4²⁵, 30, F₄, 17) (dual of [30, 5, 18]-code), using
      - (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
        linear OA(4⁶, 10, F₄, 5) (dual of [10, 4, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁶, 12, F₄, 5) (dual of [12, 6, 6]-code), using
        extended quadratic residue code Q_e(12,4) [i]
        linear OA(4⁹, 10, F₄, 8) (dual of [10, 1, 9]-code), using
        strength reduction [i] based on linear OA(4⁹, 10, F₄, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,4)), using
        dual of repetition code with length 10 [i]
        linear OA(4¹⁰, 10, F₄, 10) (dual of [10, 0, 11]-code or 10-arc in PG(9,4)), using
        complete OA / dual of code with only one code word [i]
    2. linear OA(4²³⁵, 262145, F₄, 35) (dual of [262145, 261910, 36]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]

(260−35, 260, large)-Net in Base 4 — Upper bound on s

There is no (225, 260, large)-net in base 4, because

33 times m-reduction [i] would yield (225, 227, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (260−35, 260, s)-Nets in Base 4

(260−35, 260, 15442)-Net over F4 — Constructive and digital

(260−35, 260, 233046)-Net over F4 — Digital

(260−35, 260, large)-Net in Base 4 — Upper bound on s

(260−35, 260, 15442)-Net over F₄ — Constructive and digital

(260−35, 260, 233046)-Net over F₄ — Digital