MinT - Best Known (147−12, 147, s)-Nets in Base 5

Best Known (147−12, 147, s)-Nets in Base 5

(147−12, 147, 2926412)-Net over F₅ — Constructive and digital

Digital (135, 147, 2926412)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (29, 35, 130212)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5³⁵, 130212, F₅, 6, 6) (dual of [(130212, 6), 781237, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(5³⁵, 390636, F₅, 6) (dual of [390636, 390601, 7]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(5³⁴, 390635, F₅, 6) (dual of [390635, 390601, 7]-code), using
        constructionÂ X4 applied to C_e(5) âŠ‚ C_e(3) [i] based on
        linear OA(5³³, 390625, F₅, 6) (dual of [390625, 390592, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(5²⁵, 390625, F₅, 4) (dual of [390625, 390600, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(5⁹, 10, F₅, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,5)), using
        dual of repetition code with length 10 [i]
        linear OA(5¹, 10, F₅, 1) (dual of [10, 9, 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 (100, 112, 2796200)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹¹², 2796200, F₅, 14, 12) (dual of [(2796200, 14), 39146688, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(5¹¹², 8388601, F₅, 2, 12) (dual of [(8388601, 2), 16777090, 13]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(5¹¹², 8388602, F₅, 2, 12) (dual of [(8388602, 2), 16777092, 13]-NRT-code), using
        trace code [i] based on linear OOA(25⁵⁶, 4194301, F₂₅, 2, 12) (dual of [(4194301, 2), 8388546, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25⁵⁶, 8388602, F₂₅, 12) (dual of [8388602, 8388546, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁵⁶, large, F₂₅, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(147−12, 147, large)-Net over F₅ — Digital

Digital (135, 147, large)-net over F₅, using

5¹ times duplication [i] based on digital (134, 146, large)-net over F₅, using
- t-expansion [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]

(147−12, 147, large)-Net in Base 5 — Upper bound on s

There is no (135, 147, large)-net in base 5, because

10 times m-reduction [i] would yield (135, 137, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (147−12, 147, s)-Nets in Base 5

(147−12, 147, 2926412)-Net over F5 — Constructive and digital

(147−12, 147, large)-Net over F5 — Digital

(147−12, 147, large)-Net in Base 5 — Upper bound on s

(147−12, 147, 2926412)-Net over F₅ — Constructive and digital

(147−12, 147, large)-Net over F₅ — Digital