Knuth defines a figure of merit, which describes how close the separation ${\displaystyle 1/\nu _{t}}$  is to the theoretical minimum. Under Steele & Vigna's re-notation, for a dimension ${\displaystyle d}$ , the figure ${\displaystyle f_{d}}$  is defined as:^[7]: 3

${\displaystyle f_{d}(m,a)=\nu _{d}/(\gamma _{d}^{\tfrac {1}{2}}{\sqrt[{d}]{m}})}$ ,

where ${\displaystyle a,m,\nu _{d}}$  are defined as before, and ${\displaystyle \gamma _{d}}$  being the Hermite constant of dimension d. ${\displaystyle \gamma _{d}^{\tfrac {1}{2}}{\sqrt[{d}]{m}}}$  is the smallest possible interplane separation.^[7]: 3

L'Ecuyer 1991 further introduces two measures corresponding to the minimum of ${\displaystyle f_{d}}$  across a number of dimensions.^[8] Again under re-notation, ${\displaystyle {\mathcal {M}}_{d}^{+}(m,a)}$  is the minimum ${\displaystyle f_{d}}$  for a LCG from dimensions 2 to ${\displaystyle d}$ , and ${\displaystyle {\mathcal {M}}_{d}^{*}(m,a)}$  is the same for a multiplicative congruential pseudorandom number generator (MCG), i.e. one where only multiplication is used, or ${\displaystyle c=0}$ . Steele & Vigna note that the ${\displaystyle f_{d}}$  is calculated differently in these two cases, necessitating separate values.^[7]: 13 They further define a "harmonic" weighted average figure of merit, ${\displaystyle {\mathcal {H}}_{d}^{+}(m,a)}$  (and ${\displaystyle {\mathcal {H}}_{d}^{*}(m,a)}$ ).^[7]: 13

A small variant of the infamous RANDU, with ${\displaystyle x_{n+1}=65539\,x_{n}{\bmod {2}}^{29}}$  has:^{[4]: (Table 1)}

The aggregate figures of merit are: ${\displaystyle {\mathcal {M}}_{8}^{*}(65539,2^{29})=0.018902}$ , ${\displaystyle {\mathcal {H}}_{8}^{*}(65539,2^{29})=0.330886}$ .^[a]

George Marsaglia (1972) considers ${\displaystyle x_{n+1}=69069\,x_{n}{\bmod {2}}^{32}}$  as "a candidate for the best of all multipliers" because it is easy to remember, and has particularly large spectral test numbers.^[9]

The aggregate figures of merit are: ${\displaystyle {\mathcal {M}}_{8}^{*}(69069,2^{32})=0.313127}$ , ${\displaystyle {\mathcal {H}}_{8}^{*}(69069,2^{32})=0.449578}$ .^[a]

Steele & Vigna (2020) provide the multipliers with the highest aggregate figures of merit for many choices of m = 2ⁿ and a given bit-length of a. They also provide the individual ${\displaystyle f_{d}}$  values and a software package for calculating these values.^{[7]: 14–5} For example, they report that the best 17-bit a for m = 2³² is:

• For an LCG (c ≠ 0), 0x1dab5 (121525). ${\displaystyle {\mathcal {M}}_{8}^{+}=0.6403}$ , ${\displaystyle {\mathcal {H}}_{8}^{+}=0.6588}$ .^[7]: 14
• For an MCG (c = 0), 0x1e92d (125229). ${\displaystyle {\mathcal {M}}_{8}^{*}=0.6623}$ , ${\displaystyle {\mathcal {H}}_{8}^{*}=0.7497}$ .^[7]: 14

• Entacher, Karl (January 1998). "Bad subsequences of well-known linear congruential pseudorandom number generators". ACM Transactions on Modeling and Computer Simulation. 8 (1): 61–70. doi:10.1145/272991.273009. – lists the ${\displaystyle f_{d}}$  (notated as ${\displaystyle S_{s}}$  in this text) of many well-known LCGs
  • An expanded version of this work is available as: Entacher, Karl (2001). "A Collection of Selected Pseudorandom Number Generators With Linear Structures - extended version".