MinT - Best Known (123−11, 123, s)-Nets in Base 5

Best Known (123−11, 123, s)-Nets in Base 5

(123−11, 123, 3357843)-Net over F₅ — Constructive and digital

Digital (112, 123, 3357843)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (16, 21, 2403)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5²¹, 2403, F₅, 6, 5) (dual of [(2403, 6), 14397, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(5²¹, 2404, F₅, 2, 5) (dual of [(2404, 2), 4787, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(5¹, 601, F₅, 2, 1) (dual of [(601, 2), 1201, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(5¹, s, F₅, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(5¹, 601, F₅, 2, 1) (dual of [(601, 2), 1201, 2]-NRT-code) (see above)
        linear OOA(5⁵, 601, F₅, 2, 2) (dual of [(601, 2), 1197, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(5⁵, 781, F₅, 2, 2) (dual of [(781, 2), 1557, 3]-NRT-code), using
        appending kth column [i] based on linear OA(5⁵, 781, F₅, 2) (dual of [781, 776, 3]-code), using
        Hamming code H(5,5) [i]
        linear OOA(5¹⁴, 601, F₅, 2, 5) (dual of [(601, 2), 1188, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(5¹⁴, 1202, F₅, 5) (dual of [1202, 1188, 6]-code), using
        trace code [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (91, 102, 3355440)-net over F₅, using
  - trace code for nets [i] based on digital (40, 51, 1677720)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁵¹, 1677720, F₂₅, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(25⁵¹, 8388601, F₂₅, 11) (dual of [8388601, 8388550, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁵¹, large, F₂₅, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(123−11, 123, large)-Net over F₅ — Digital

Digital (112, 123, large)-net over F₅, using

4 times m-reduction [i] based on digital (112, 127, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²⁷, large, F₅, 15) (dual of [large, large−127, 16]-code), using
  - 7 times code embedding in larger space [i] based on linear OA(5¹²⁰, large, F₅, 15) (dual of [large, large−120, 16]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(123−11, 123, large)-Net in Base 5 — Upper bound on s

There is no (112, 123, large)-net in base 5, because

9 times m-reduction [i] would yield (112, 114, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (123−11, 123, s)-Nets in Base 5

(123−11, 123, 3357843)-Net over F5 — Constructive and digital

(123−11, 123, large)-Net over F5 — Digital

(123−11, 123, large)-Net in Base 5 — Upper bound on s

(123−11, 123, 3357843)-Net over F₅ — Constructive and digital

(123−11, 123, large)-Net over F₅ — Digital