MinT - Best Known (54, 54+16, s)-Nets in Base 5

Best Known (54, 54+16, s)-Nets in Base 5

(54, 54+16, 400)-Net over F₅ — Constructive and digital

Digital (54, 70, 400)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 10)-net over F₅, using
  - net from sequence [i] based on digital (1, 9)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 1 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (45, 61, 390)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁶¹, 390, F₅, 16, 16) (dual of [(390, 16), 6179, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(5⁶¹, 3120, F₅, 16) (dual of [3120, 3059, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁶¹, 3125, F₅, 16) (dual of [3125, 3064, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(54, 54+16, 3376)-Net over F₅ — Digital

Digital (54, 70, 3376)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷⁰, 3376, F₅, 16) (dual of [3376, 3306, 17]-code), using
- 242 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 17 times 0, 1, 35 times 0, 1, 64 times 0, 1, 107 times 0) [i] based on linear OA(5⁶¹, 3125, F₅, 16) (dual of [3125, 3064, 17]-code), using
  - an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(54, 54+16, 1229180)-Net in Base 5 — Upper bound on s

There is no (54, 70, 1229181)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 8 470355 960725 875028 098601 120791 342881 125753 280225 > 5⁷⁰ [i]