MinT - Best Known (141−13, 141, s)-Nets in Base 5

Best Known (141−13, 141, s)-Nets in Base 5

(141−13, 141, 2796410)-Net over F₅ — Constructive and digital

Digital (128, 141, 2796410)-net over F₅, using

5¹ times duplication [i] based on digital (127, 140, 2796410)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (12, 18, 210)-net over F₅, using
    - net defined by OOA [i] based on linear OOA(5¹⁸, 210, F₅, 6, 6) (dual of [(210, 6), 1242, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(5¹⁸, 630, F₅, 6) (dual of [630, 612, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(3) [i] based on
        linear OA(5¹⁷, 625, F₅, 6) (dual of [625, 608, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(5¹³, 625, F₅, 4) (dual of [625, 612, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(5¹, 5, F₅, 1) (dual of [5, 4, 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]

(141−13, 141, large)-Net over F₅ — Digital

Digital (128, 141, large)-net over F₅, using

5⁴ times duplication [i] based on digital (124, 137, large)-net over F₅, using
- t-expansion [i] based on digital (121, 137, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹³⁷, large, F₅, 16) (dual of [large, large−137, 17]-code), using
    - 16 times code embedding in larger space [i] based on linear OA(5¹²¹, large, F₅, 16) (dual of [large, large−121, 17]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(141−13, 141, large)-Net in Base 5 — Upper bound on s

There is no (128, 141, large)-net in base 5, because

11 times m-reduction [i] would yield (128, 130, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (141−13, 141, s)-Nets in Base 5

(141−13, 141, 2796410)-Net over F5 — Constructive and digital

(141−13, 141, large)-Net over F5 — Digital

(141−13, 141, large)-Net in Base 5 — Upper bound on s

(141−13, 141, 2796410)-Net over F₅ — Constructive and digital

(141−13, 141, large)-Net over F₅ — Digital