MinT - Best Known (123, 123+13, s)-Nets in Base 5

Best Known (123, 123+13, s)-Nets in Base 5

(123, 123+13, 2796259)-Net over F₅ — Constructive and digital

Digital (123, 136, 2796259)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (8, 14, 59)-net over F₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 26)-net over F₅, using
      - net defined by OOA [i] based on linear OOA(5⁴, 26, F₅, 3, 3) (dual of [(26, 3), 74, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(5⁴, 26, F₅, 2, 3) (dual of [(26, 2), 48, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5⁴, 26, F₅, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
        ovoid in PG(3,Â 5) [i]
    2. digital (4, 10, 33)-net over F₅, using
      - linear code embedding found in a computer search [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]

(123, 123+13, large)-Net over F₅ — Digital

Digital (123, 136, large)-net over F₅, using

t-expansion [i] based on digital (121, 136, large)-net over F₅, using
- 1 times m-reduction [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]

(123, 123+13, large)-Net in Base 5 — Upper bound on s

There is no (123, 136, large)-net in base 5, because

11 times m-reduction [i] would yield (123, 125, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (123, 123+13, s)-Nets in Base 5

(123, 123+13, 2796259)-Net over F5 — Constructive and digital

(123, 123+13, large)-Net over F5 — Digital

(123, 123+13, large)-Net in Base 5 — Upper bound on s

(123, 123+13, 2796259)-Net over F₅ — Constructive and digital

(123, 123+13, large)-Net over F₅ — Digital