MinT - Best Known (144−13, 144, s)-Nets in Base 5

Best Known (144−13, 144, s)-Nets in Base 5

(144−13, 144, 2797244)-Net over F₅ — Constructive and digital

Digital (131, 144, 2797244)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (16, 22, 1044)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5²², 1044, F₅, 6, 6) (dual of [(1044, 6), 6242, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(5²², 3132, F₅, 6) (dual of [3132, 3110, 7]-code), using
      - constructionÂ X4 applied to C_e(5) âŠ‚ C_e(3) [i] based on
        linear OA(5²¹, 3125, F₅, 6) (dual of [3125, 3104, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(5¹⁶, 3125, F₅, 4) (dual of [3125, 3109, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(5⁶, 7, F₅, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,5)), using
        dual of repetition code with length 7 [i]
        linear OA(5¹, 7, F₅, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. digital (109, 122, 2796200)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹²², 2796200, F₅, 14, 13) (dual of [(2796200, 14), 39146678, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(5¹²², 8388601, F₅, 2, 13) (dual of [(8388601, 2), 16777080, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(5¹²², 8388602, F₅, 2, 13) (dual of [(8388602, 2), 16777082, 14]-NRT-code), using
        trace code [i] based on linear OOA(25⁶¹, 4194301, F₂₅, 2, 13) (dual of [(4194301, 2), 8388541, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25⁶¹, 8388602, F₂₅, 13) (dual of [8388602, 8388541, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁶¹, large, F₂₅, 13) (dual of [large, large−61, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(144−13, 144, large)-Net over F₅ — Digital

Digital (131, 144, large)-net over F₅, using

t-expansion [i] based on digital (129, 144, large)-net over F₅, using
- 2 times m-reduction [i] based on digital (129, 146, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴⁶, large, F₅, 17) (dual of [large, large−146, 18]-code), using
    - 15 times code embedding in larger space [i] based on linear OA(5¹³¹, large, F₅, 17) (dual of [large, large−131, 18]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(144−13, 144, large)-Net in Base 5 — Upper bound on s

There is no (131, 144, large)-net in base 5, because

11 times m-reduction [i] would yield (131, 133, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (144−13, 144, s)-Nets in Base 5

(144−13, 144, 2797244)-Net over F5 — Constructive and digital

(144−13, 144, large)-Net over F5 — Digital

(144−13, 144, large)-Net in Base 5 — Upper bound on s

(144−13, 144, 2797244)-Net over F₅ — Constructive and digital

(144−13, 144, large)-Net over F₅ — Digital