Some Results on the Weight Spectrum of Binary Reed-Muller Codes $RM(m-7,m)$

1 Introduction

• A complete determination of W(m−7, m) remains open, and the paper presents its result as a constructive inclusion rather than a full spectrum classification.The introduction also notes contemporaneous work on RM(7, 14), whose outstanding targets included the four low weights left unresolved here.Source: §1 · pp. 2–3
• For every m ≥14, the construction covers all even weights in [336, 2^m−336], the listed low weights, and their complements.The low-weight set is {304, 308, 312, 314, 316, 318, 320, 324, 328, 332}.Source: §1 · p. 2
• The base inclusion in RM(7, 14) combines four-block concatenation and degree-six monomial witnesses with spectrum inputs from RM(5, 10) and RM(6, 12).A concatenation-based induction lifts the base inclusion to every m ≥14.Source: §1 · pp. 2–3
• The remaining low base-code targets are 322, 326, 330, and 334; the paper analyzes them in several restricted construction families.The exclusions include four-block branches, selected corrections in a fixed Carlet-type Maiorana–McFarland shell, and sparse degree-seven coordinate-monomial sums.Source: §1 · p. 3

2.1 Boolean functions and Reed–Muller codes

• RM(r, m) consists of truth tables of m-variable Boolean functions of algebraic degree at most r, and its weight spectrum is the set of weights occurring in the code.The paper studies the spectrum as a set, not the full weight distribution.Source: §2.1 · p. 4

2.2 Standard Reed–Muller facts

• The minimum nonzero weight is 2^(m−r), every weight is divisible by 2^⌊(m−1)/r⌋, and the spectrum is symmetric under w ↔ 2^m−w.Minimum-weight codewords are indicators of affine (m−r)-flats; invertible affine changes preserve degree and weight.Source: §2.2 · p. 4

2.3 Lifting and concatenation

• Lifting a codeword to m+t variables multiplies its weight by 2^t; concatenation combines functions into blocks, with a four-block order specified for two added variables.The degree calculation for the four-block construction used in the main argument is given separately in Lemma 3.1.Source: §2.3 · p. 4

3 A constructive inclusion for the weight spectrum of RM(m −7, m)

• The RM(7,14) base construction combines with known RM(5,10) and RM(6,12) spectrum inclusions to form a contiguous interval that is lifted to all m ≥ 14.The base weights are constructed using four-block concatenation and explicit degree-6 monomial witnesses.Source: §3 · p. 5

3.1 The four-block construction and monomial weight formulae

• Lemma 3.1 states that for g0, g3 ∈ RM(5,12) and g1, g2 ∈ RM(6,12), the concatenation g0 ∥ g1 ∥ g2 ∥ (g1 + g2 + g3) belongs to RM(7,14).Given these degree bounds, every term has degree at most 7.Source: §3.1 · p. 5

3.2 Base weights in RM(7, 14)

• Flipping selected nonzero blocks attains weights congruent to 2 (mod 4) in [314, 318] ∪ [446, 450], [4098, 4102] ∪ [4238, 4250] ∪ [4354, 4362].The latter intervals follow from complementing one of the three nonzero blocks in the four-block construction.Source: §3.2 · pp. 7–8
• Four-degree-6-monomial constructions attain every multiple of 4 in [4380, 4384] and weights congruent to 2 (mod 4) in {4382} ∪ [4390, 4394].The constructions use selected monomials in the nonzero blocks and complement the g1 block.Source: §3.2 · p. 9
• Constructions 3.6, 3.10, and 3.13 collectively attain every integer congruent to 2 (mod 4) in [338, 474] ∪ [538, 542].These are base-code weights in RM(7,14).Source: §3.2 · p. 10
• Explicit witnesses and Table 1 cover every weight congruent to 2 (mod 4) in [338, 474] ∪ [538, 542].The displayed witnesses include 338, 354, 474, 538, and 542; Table 1 supplies the remaining weights in those ranges.Source: §3.2 · pp. 10–11

3.3 Base interval completion and lifting

• Concatenating RM(5, 12) blocks with base constructions fills the 2 (mod 4) ranges, while doubled RM(6, 12) weights provide 0 (mod 4) coverage.Fixed shifts fill the lower endpoint and the gap between ranges; complementation then supplies the stated finite low-weight set and its complements.Source: §3.3 · p. 12
• RM(7, 14) contains every even weight in [336, 8192], combining the 2 (mod 4) and 0 (mod 4) constructions.The 2 (mod 4) construction covers [338, 8190], while the 0 (mod 4) construction covers [304, 8192].Source: §3.3 · p. 12
• For every m ≥ 14, the spectrum of RM(m −7, m) contains A, all weights 336 + 2i through 2^m − 336, and the complements of weights in A.The induction uses zero-block embedding, concatenation with a weight-336 codeword, and complementation symmetry.Source: §3.3 · pp. 12–13

4.1 Exclusions for two restricted four-block families

• The direct g3 = 0 four-block family and the weight-126 one-point-intersection minimum-weight branch contain none of the four target weights.The second exclusion allows a correction in RM(5, 12).Source: §4.1 · pp. 13–14

4.2 Affine-flat corrections to a fixed Carlet-type Maiorana–McFarland outer shell

• Within the fixed outer shell, single affine 5-flats, sums of disjoint affine 5-flats, and sums of two intersecting affine 5-flats avoid the reduced targets.The reduced target set is Tred = {17, 18, 19, 20}; the exclusions do not cover sums of three or more intersecting flats or corrections outside this shell.Source: §4.2 · pp. 15–16

4.3 Structured inner corrections in the fixed Carlet-type Maiorana–McFarland outer shell

• No correction g in RM(7, 12) within the fixed Carlet-type outer shell yields a target weight; equivalently, wt(g|Z) avoids {17, 18, 19, 20}.The exclusion is complete for inner corrections with this outer shell fixed.Source: §4.3 · p. 16
• Further fixed-shell exclusions cover specified pure-cross, low-block-rank, and normalized base-perturbation correction families.For the v-distinct pure-cross profile subfamily, the relaxed row minimum is at least 31 and is attained.Source: §4.3 · p. 17
• Single, paired, and three-core genuinely mixed affine 5-flat corrections within the fixed outer shell also avoid the reduced targets.The three-core exclusion covers every member of the family M3.Source: §4.3 · p. 17

4.4 Sparse sums of degree-7 coordinate monomials

• Sums of at most eight squarefree coordinate degree-7 monomials do not have any target weight.This exclusion does not extend to arbitrary affine 7-flat indicators or general highest-degree cosets.Source: §4.4 · p. 18
• Three broader routes—highest-degree cosets, affine 7-flat assembly beyond coordinate monomials, and global Maiorana–McFarland corepooling—remain untreated.The paper derives no additional exclusions from these routes.Source: §4.4 · p. 18

5 Conclusion

• For every m ≥14, the spectrum contains all even weights in [336, 2^m −336], the stated finite low-weight set, and their complements.The base inclusion in RM(7, 14) uses four-block concatenation and degree-six monomial witnesses, then a concatenation-based induction lifts it to all m ≥14.Source: §5 · p. 18
• The four base-code targets 322, 326, 330, and 334 were excluded only in several restricted construction families; their occurrence in RM(7, 14) remains open.The exclusions cover two four-block branches, corrections to a fixed Carlet-type outer shell, and sums of at most eight degree-7 coordinate monomials.Source: §5 · p. 19

A Exact verification and witness table for the base interval

• The appendix gives a representative exact weight calculation and records additional monomial witnesses used for the base interval.For the six-monomial construction, the intersection-count tuple is (384, 183, 64, 23, 7, 1), yielding weight 170.Source: A Exact verification and witness table… · p. 20, §A.1 · p. 20

A.2 Remaining monomial witnesses

• Each Table 1 row gives monomial witnesses for a target weight via the word 0∥g1∥g2∥(g1 + g2).Together with five witnesses displayed in the proof, the rows cover all weights congruent to 2 modulo 4 in [338, 474] ∪[538, 542].Source: §A.2 · p. 20, Table 1 · p. 21

B Detailed exclusions for restricted four-block families

• The following exclusions concern target weights T = {322, 326, 330, 334} and affine d-flats in V.The supplied passage gives this setup but no exclusion result.Source: B Detailed exclusions for restricted… · p. 21

B.1 The direct four-block family

• A weight-126 word is the symmetric difference of two affine 6-flats meeting in one point, while a weight-96 word uses two meeting in an affine 4-flat.The weight-112 forms include a two-flat type and a quadratic type; these normal forms support the intersection exclusions.Source: §B.1 · p. 22
• For a weight-126 support, intersections with an affine 6-flat cannot have size 23 or 27; for weight 112, they cannot have size 9 or 11.For a weight-96 support, the possible intersection sizes are {0, 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48}.Source: §B.1 · pp. 22–23
• No two-dimensional subspace of RM(6, 12) has its three nonzero word weights summing to 322, 326, 330, or 334.The proof enumerates possible weight triples from the low-weight spectrum and rules them out using jump and intersection bounds, including a separate check of the remaining (96, 112, 126) case.Source: §B.1 · pp. 22–24

B.2 Exclusion of the weight-126 minimum-weight branch

• In the weight-126 branch, exact enumeration of RM(5, 12) split profiles leaves only μ = 132, 136, or 140 for a target word.The filtering uses the required relation among the intersection counts and target weights 194, 198, 202, and 206.Source: §B.2 · p. 24

B.2.1 The weight-192 overlap verification

• For affine 7-flats C and D meeting in an affine 5-flat, |(C△D) ∩S| cannot be 56 or 60.The proof enumerates affine cosets and quotient configurations for each admissible direction type; neither overlap value occurs.Source: §B.2.1 · p. 25

C Punctured affine-flat geometry in the fixed shell

• The appendix sets V = V6 × V6 and Z = (V6 \ {0}) × (V6 \ {0}), with H1 = V6 × {0}, H2 = {0} × V6, and H = H1 ∪ H2, so Z = V \ H.Source: C Punctured affine-flat geometry in… · p. 25

• The size |X ∩ H| is zero, twice an integer through r, or one of two sums determined by whether the two affine pieces meet.When both pieces meet, their intersection can only be 0; when disjoint, an additional direction joining them is required.Source: §C.1 · p. 25
• |X ∩ Z| has 15 listed possibilities for r = 5 and 7 for r = 3, as derived from the classification of |X ∩ H|.For r = 5 the values are {0, 15, 16, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}; for r = 3 they are {0, 3, 4, 5, 6, 7, 8}.Source: §C.1 · p. 25

C.2 Two affine 5-flat corrections

• For distinct affine 5-flats with nonempty intersection, dimensions 0, 1, 2, and 4 cannot produce a reduced weight in Tred.If either flat has zero punctured size, the restriction is a single affine 5-flat indicator and its weight is in P5, not Tred.Source: §C.2 · pp. 25–26
• For intersection dimensions 0, 1, and 2, the reduced weights are at least 28, 26, and 22, respectively, and none lies in Tred.The bound uses wt(g|H) ≤ 34; dimension 4 is excluded because the sum of the two indicators is another affine hyperplane indicator.Source: §C.2 · p. 26
• For intersection dimension 3, any target weight in Tred forces one of the candidate triples in Table 2, where the reduced weight is a + b − 2c.The possible punctured sizes a and b come from P5, while c comes from P3.Source: §C.2 · p. 26, Table 2 · p. 26
• For punctured sizes 15, 16, and 21, the intersections with affine 3-flats have constrained sizes: respectively {0, 3, 4, 7, 8}, {0, 4, 8}, and no affine 3-flat with a constant coordinate block.The type-15 and type-16 cases have a constant coordinate block on such a 3-flat; the type-21 case has product form (U \ {0}) × (W \ {0}).Source: §C.2 · pp. 26–27
• The dimension-3 candidates are all excluded: local geometry rules out the required c values for punctured sizes 15 or 16, then contradicts the remaining (15, 21, 8) case.The final triple would require a common affine 3-flat in both punctured parts, but the type-15 side forces a constant coordinate block while the type-21 side permits none.Source: §C.2 · p. 27

D Row-profile and mixed-core analysis in the fixed shell

• The finite profile calculation is retained only for the quantitative pure-cross refinement, not for the fixed-shell exclusion.The appendix states that its target exclusions follow from explicit algebraic and geometric arguments.Source: D Row-profile and mixed-core analysis… · p. 28

D.1 Pure-cross factors and separated presentations

• For punctured local spaces Rd, the minimum nonzero weight is 26−d − 1, and the lighter of q and its complement has lower bound µd = 26−d − 1 for q in Rd \ Rd−1.For d = 1 through 5, the µd values are 31, 15, 7, 3, and 1; positive weights in R2 are {15, 16, 23, 24, 27, 28, 31}.Source: §D.1 · p. 28
• A degree-compatible separated presentation writes P as a sum of products αi(u)qi(v) with ai + bi ≤ 7, and every restriction from RM(7, 12) admits this form.Terms depending only on u are absorbed into h; a degree-six v-monomial on U can be replaced by degree at most five.Source: §D.1 · p. 29
• No word h + P in the degree-compatible separated family has weight in Tred; this also excludes L2, L3, the pure-cross family, and sums of at most three affine 5-flat indicators.The result applies when h is arbitrary and P has total degree at most 7.Source: §D.1 · p. 29
• The low-weight row argument excludes Tred across degree-compatible separated presentations, with its final degree-five case forcing wt(g) ≥ 22.For top-layer rank at least two, the proof obtains a lower bound of 23; rank one is handled by the Hamming-code coset structure and lower-degree row contributions.Source: §D.1 · pp. 29–31
• For the pure-cross family C≤5, no h + P has weight in Tred; in the v-distinct profile subfamily, the minimum is 31.The refinement checks profiles through five summands, and equality is attained by a specified one-row construction.Source: §D.1 · pp. 31–32

D.2 Mixed affine cores

• A genuinely mixed affine core has no reduced weight in Tred; its possible values not exceeding 24 are contained in {15, 16, 22, 23, 24}.The formula for |S ∩ Z| depends on the dimensions of the linear part's intersections with H1 and H2 and whether S meets those subspaces or contains 0.Source: §D.2 · p. 32

D.3 Three-core corrections

• For sizes (22, 23, 23), a symmetric-difference size of 20 forces all three pairwise intersections to have size 8 and the triple intersection to be empty.This follows from the conditional arithmetic in Lemma D.5.Source: §D.3 · p. 33
• For core sizes (22, 22, 24), the required value 25 of a12+a13+a23−2t cannot occur.If all pairwise affine intersections have dimension at most 3, the expression is at most 24; a dimension-4 intersection instead reduces the correction to an excluded case or cancellation.Source: §D.3 · p. 33

E Profile analysis of sparse sums of degree-7 coordinate monomials

• In the sparse-monomial setup, the target set is {322, 326, 330, 334}, and sums are over F2.Each degree-7 coordinate monomial has a support set A_I of size 128.Source: E Profile analysis of sparse sums of… · p. 33

E.1 The symmetric-difference weight formula

• The support of a sum of coordinate monomials is the symmetric difference of their individual support sets.This representation underlies the symmetric-difference weight formula.Source: §E.1 · p. 34

E.2 Exact low-term analysis for 0 ≤s ≤4

• For 0–4 distinct monomials, the resulting weights do not intersect T = {322, 326, 330, 334}.The two-monomial weights are {128, 192, 224, 240, 248, 252, 254}; the three- and four-term cases use feasible Venn-region profiles.Source: §E.2 · pp. 34–35

E.3 Profile enumeration for 5 ≤s ≤7

• For 5–7 distinct monomials, enumerating feasible Venn-region profiles yields no weight in T = {322, 326, 330, 334}.The profiles determine the weights and account for coordinate-permutation orbits, with summand permutations identifying unordered families.Source: §E.3 · p. 35

E.4 Complementary-pair reduction for s = 8

• For eight distinct monomials, every target-weight configuration contains a complementary pair, and the fixed-pair profile enumeration excludes all four target weights.The pair can be normalized to the two complementary 7-subsets {1,…,7} and {8,…,14}.Source: §E.4 · pp. 35–36

E.5 Repeated summands

• Repeated monomials cancel in pairs over F2, so any repeated-summand expression reduces to at most eight distinct monomials; the exclusion is confined to squarefree degree-7 coordinate monomial sums.Combining the exclusions for up to eight distinct summands rules out T within this family.Source: §E.5 · p. 36