Completed Maiorana–McFarland Class

• The paper establishes that the completed Maiorana–McFarland class is the EA-invariant closure of classical MM bent functions, unifying several constructions.
• It characterizes membership via derivative tests where a bent function belongs if there exists an m-dimensional subspace on which all second derivatives vanish.
• The work examines permutation properties (P1/P2), concatenation methods, and asymptotic count comparisons to delineate the class’s boundaries.

The completed Maiorana–McFarland class is the EA-invariant enlargement of one of the central primary bent-function constructions. In the Boolean setting on $n=2m$ variables, the classical Maiorana–McFarland class consists of functions

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$

with $\pi$ a permutation of $\mathbb F_2^m$ and $h$ an arbitrary Boolean function; the completed class, usually written $\mathcal M^\#$, is the smallest EA-invariant class containing $\mathcal M$ (Kudin et al., 19 Aug 2025). A standard equivalent characterization is Dillon’s second-derivative criterion: a bent $f\in\mathcal B_{2m}$ belongs to $\mathcal M^\#$ if and only if there exists an $m$-dimensional linear subspace $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$0 such that

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$1

In later literature the same phrase also appears in $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$2-ary and vectorial settings, sometimes as an EA-completed class and sometimes as a secondary “completed/generalized” MM construction; the distinction is substantive rather than terminological (Qi et al., 2015).

1. Classical definition and EA-completion

In the strict Boolean form, the Maiorana–McFarland class is

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$3

Its completed version is the extended-affine closure

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$4

Thus $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$5 is not another explicit normal form; it is the equivalence-stable class generated by $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$6 (Kudin et al., 19 Aug 2025).

This distinction becomes important as soon as one leaves the balanced $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$7 split. A recent “almost Maiorana–McFarland” family studies

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$8

as a strict subfamily of the generalized Maiorana–McFarland class $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$9. These functions retain the MM pattern of affine restrictions in one block of variables, but they are not members of the strict MM class $\pi$0 unless additional structure is present (Kudin et al., 19 Aug 2025).

The completed class therefore plays two roles at once. It is an equivalence-stable replacement for the rigid explicit template $\pi$1, and it is the benchmark against which broader MM-like constructions are tested. Much of the recent literature is concerned less with constructing members of $\pi$2 than with deciding whether a bent function that “looks MM-like” is actually inside $\pi$3.

2. Derivative characterization and $\pi$4-subspaces

The derivative formulation of $\pi$5 is encoded by $\pi$6-subspaces. For a Boolean function $\pi$7, the first derivative is

$\pi$8

and the second derivative is

$\pi$9

A vector subspace $\mathbb F_2^m$0 is an $\mathbb F_2^m$1-subspace of $\mathbb F_2^m$2 if

$\mathbb F_2^m$3

If $\mathbb F_2^m$4, Dillon’s criterion becomes: $\mathbb F_2^m$5 For an MM function

$\mathbb F_2^m$6

the canonical example is the $\mathbb F_2^m$7-dimensional subspace

$\mathbb F_2^m$8

which is always an $\mathbb F_2^m$9-subspace (Pasalic et al., 2023).

This viewpoint supplies an EA-invariant. The number of $h$0-subspaces of a Boolean function is invariant under equivalence, so two bent functions with different numbers of $h$1-subspaces are inequivalent. This turns second-order derivatives into a classification device, not merely a membership test (Pasalic et al., 2023).

A plausible implication is that $h$2 is best viewed geometrically. What matters is not only the explicit form $h$3, but the existence, dimension, and arrangement of vanishing second-derivative subspaces. That is exactly why the same class reappears in concatenation arguments, exclusion criteria, and low-linearity constructions.

3. Internal structure of $h$4: uniqueness, $h$5, $h$6, and linearity index

A central refinement concerns how many maximal $h$7-subspaces an MM bent function possesses. For

$h$8

every MM function has the canonical $h$9-dimensional $\mathcal M^\#$0-subspace

$\mathcal M^\#$1

but this need not be unique. The literature isolates two permutation properties controlling uniqueness. The stronger condition is

$\mathcal M^\#$2

If $\mathcal M^\#$3 satisfies $\mathcal M^\#$4, then $\mathcal M^\#$5 has no linear structures and $\mathcal M^\#$6 is the only $\mathcal M^\#$7-dimensional $\mathcal M^\#$8-subspace of $\mathcal M^\#$9. A weaker condition $\mathcal M$0 is defined through subspaces $\mathcal M$1 with

$\mathcal M$2

and excludes the existence of a complementary dual collapse

$\mathcal M$3

For non-affine $\mathcal M$4, $\mathcal M$5 is equivalent to uniqueness of the maximal $\mathcal M$6-subspace for $\mathcal M$7. The paper also gives the equivalent reformulation

$\mathcal M$8

and proves

$\mathcal M$9

with strict implication: among the 75 equivalence classes of quadratic permutations on $f\in\mathcal B_{2m}$0, 34 satisfy $f\in\mathcal B_{2m}$1, while only 2 satisfy $f\in\mathcal B_{2m}$2 (Kudin et al., 19 Aug 2025).

The same paper organizes this geometry by the linearity index

$f\in\mathcal B_{2m}$3

For bent functions on $f\in\mathcal B_{2m}$4 variables,

$f\in\mathcal B_{2m}$5

and

$f\in\mathcal B_{2m}$6

This makes $f\in\mathcal B_{2m}$7 the maximal-linearity extreme. The opposite extreme is the class of $f\in\mathcal B_{2m}$8-optimal bent functions, defined by

$f\in\mathcal B_{2m}$9

These are described as opposite to Maiorana–McFarland bent functions because MM and $\mathcal M^\#$0 functions have maximal possible linearity index $\mathcal M^\#$1, whereas $\mathcal M^\#$2-optimal functions have the minimum possible index (Kudin et al., 19 Aug 2025).

4. Generalized and almost MM forms at the boundary of $\mathcal M^\#$3

The almost-MM family

$\mathcal M^\#$4

gives a precise test case for how far MM flavor can be pushed before one exits $\mathcal M^\#$5. Its bentness is completely characterized: $\mathcal M^\#$6 is bent if and only if the fibers

$\mathcal M^\#$7

form a partition of $\mathcal M^\#$8 into $\mathcal M^\#$9-dimensional affine subspaces, and the restriction of $m$0 to each fiber has odd Hamming weight, equivalently weight $m$1 or $m$2. For a fixed $m$3 with this partition property, the number of valid $m$4 is

$m$5

The dual is also determined fiberwise (Kudin et al., 19 Aug 2025).

Membership in $m$6 is subtler. A sufficient condition is the existence of a nonzero $m$7 such that every affine plane $m$8 contains the common direction $m$9; then

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$00

is an $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$01-dimensional $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$02-subspace, so $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$03. Partitions with such a common direction are called non-proper, and those without it are called proper. The paper proves that non-proper partitions force $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$04, but the converse fails: the condition is sufficient, not necessary (Kudin et al., 19 Aug 2025).

The same work also identifies when an almost-MM bent function is actually in the strict MM class. A bent $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$05 belongs to $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$06 if and only if it can be written in the almost-MM form with $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$07 depending only on the last $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$08 variables and

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$09

balanced in such a way that $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$10 extends to a permutation of $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$11 (Kudin et al., 19 Aug 2025).

The 8-variable case is especially explicit. The paper found exactly $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$12 decompositions of $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$13 into $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$14-dimensional affine subspaces; among them $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$15 are proper. For these $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$16 proper partitions, all obtained bent functions were reported to be outside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$17. The resulting lower bound is at least

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$18

distinct 8-variable bent functions outside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$19, compared with an estimated total of approximately

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$20

8-variable bent functions inside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$21 (Kudin et al., 19 Aug 2025).

5. Outside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$22: concatenation, cubic counterexamples, and asymptotic limits

The most systematic Boolean exclusion machinery uses $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$23-subspaces. For bent 4-concatenation

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$24

if $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$25, then there must exist a common $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$26-dimensional $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$27-subspace in the four constituents. This yields generic non-membership criteria. In particular, suitably chosen MM components can be concatenated to produce bent functions outside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$28 for any even $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$29; explicit 8-variable examples constructed this way are also outside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$30 (Pasalic et al., 2023).

A complementary MM-based route uses the dual bent condition. For MM components

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$31

if the permutations satisfy the $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$32 property and the offsets satisfy

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$33

then the 4-concatenation is bent. Under generic hypotheses including property $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$34 for $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$35 and $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$36, together with the absence of linear structures in the components of $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$37, the resulting concatenation is outside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$38 (Polujan et al., 2023).

The completed MM class is also known not to exhaust cubic bent functions. All cubic bent functions in 6 and 8 variables belong to $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$39, but this fails in larger even dimensions. There exist cubic bent functions outside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$40 for all even $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$41; moreover, homogeneous cubic bent functions outside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$42 exist for all $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$43, and homogeneous cubic bent functions without affine derivatives exist outside $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$44 for all $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$45 (Polujan et al., 2019).

Not every proposed route away from MM is genuinely productive. A corrective note on Rothaus-based iteration proved that the condition

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$46

has only the trivial solution

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$47

so the iterated Rothaus construction collapses to

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$48

The paper concludes that this iterative Rothaus mechanism contributes no genuinely new bent functions and does not provide a nontrivial route beyond MM or its completion (Guo et al., 14 Feb 2025).

From a counting perspective, the completed MM family is asymptotically small relative to broader MM-inspired generalizations. A counting paper estimates the leading scale of the completed MM family by

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$49

in $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$50-asymptotics, while a modification of the MM family based on ordered partitions of $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$51 into affine $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$52-subspaces yields

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$53

That paper does not identify its construction with $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$54, but it does show that affine-subspace generalizations of the MM philosophy have much larger asymptotic growth than the classical completed class (Potapov et al., 2021).

6. $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$55-ary, vectorial, and alternate usages of “completed MM”

Outside the binary Boolean setting, the phrase “completed Maiorana–McFarland class” is used in several adjacent but nonidentical ways. The following summary captures the main strands.

Strand	Representative form	Relation to completed MM
$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$56-ary EA-completed class	$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$57	Completed class = all EA-equivalent functions (Helleseth et al., 28 Jul 2025)
Weakly regular $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$58-ary completion principle	MM base plus $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$59	“Completed/generalized MM” means trace-augmentation of an MM base (Qi et al., 2015)
Vectorial $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$60-ary secondary construction	$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$61	MM supplies the core $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$62 satisfying $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$63 (Bapić, 2022)
Vectorial binary maximal bent components	MM-derived components with derivative test	Some resulting components are outside the complete MM class (Xie et al., 2023)

In the ternary setting, one paper defines completion exactly as EA-closure: a class is complete if it is a union of EA-equivalence classes, and the completed class of $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$64 is the smallest complete class containing it. It then proves a derivative criterion: if there exists an $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$65-dimensional subspace $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$66 such that

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$67

and every nonzero first derivative $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$68 with $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$69 is balanced, then $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$70 is a regular bent function in the completed MM class. This criterion is used to construct ternary degree-4 binomial and trinomial families inside the completed MM class (Helleseth et al., 28 Jul 2025).

A different use of “completed/generalized MM” occurs in the $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$71-ary weakly regular bent literature. Starting from the MM base

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$72

with $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$73 a linearized permutation polynomial, the paper adds an arbitrary reduced polynomial in trace coordinates: $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$74 If the trace-orthogonality condition

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$75

holds, then $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$76 remains weakly regular bent. Here “completion” refers to a secondary trace-polynomial augmentation of an MM bent function, not to an EA-closure (Qi et al., 2015).

In vectorial $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$77-ary work, the classical MM family

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$78

serves as the base object in a $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$79-driven secondary construction. If the component duals satisfy

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$80

then

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$81

with trace-coordinate perturbation $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$82 is again vectorial $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$83-ary weakly regular bent; this yields new infinite families from the $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$84-ary MM class without defining a separate completed class (Bapić, 2022).

Finally, in the vectorial binary setting of maximal bent components, the phrase “complete Maiorana–McFarland class” is tied to a second-order derivative condition. Starting from MM bent components

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$85

a secondary construction produces new vectorial functions with $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$86 bent components. For the explicit family

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$87

the paper computes

$$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$88

and shows that the complete-MM derivative criterion fails, so the constructed components are outside the complete Maiorana–McFarland class (Xie et al., 2023).

Taken together, these strands show that “completed Maiorana–McFarland class” has a stable Boolean-EA meaning, but its extensions split into at least three non-equivalent ideas: EA-completion, trace-coordinate completion of a fixed MM core, and derivative-based tests for inclusion or exclusion in vectorial and $$f(x,y)=x\cdot \pi(y)+h(y),\qquad x,y\in \mathbb F_2^m,$$89-ary settings.