Kim’s Building-Up Construction in Self-Dual Codes

• Kim’s building-up construction is a method to generate longer self-dual codes from shorter ones by adjoining controlled hyperbolic pieces and compensation terms.
• It applies to binary codes (extending length from 2n to 2n+2) and to odd-characteristic cases (extending from 2n to 2n+4) using bilinear forms and isotropic lines.
• The algorithmic approach has produced new extremal codes and established formal equivalence with Hilbert-symbol methods through rigorous algebraic and computational frameworks.

Kim’s building-up construction is a predecessor–successor method for producing self-dual codes from shorter self-dual codes by adjoining a controlled hyperbolic piece and compensating correction terms. In the binary formulation, it starts from a self-dual code of length $2n$ and produces a self-dual code of length $2n+2$; in the odd-characteristic and ring-theoretic formulation completed by Lee and Kim, it starts from length $2n$ and produces length $2n+4$. The construction is accompanied by converse statements: longer self-dual codes can be reduced to shorter ones by removing the added coordinates, and a later formalization identifies the binary construction with Chinburg–Zhang’s Hilbert-symbol construction through isotropic lines in a hyperbolic plane (Lee et al., 2012, Baek et al., 9 Apr 2026).

1. Core schema and ambient hypotheses

A linear code $C \subset \F_q^n$ is self-dual if

$C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$

where $(u,v)=\sum_{i=1}^n u_i v_i$ is the standard Euclidean inner product. In the ring-theoretic setting used by Lee–Kim, the same construction is organized around the corresponding bilinear form over a finite chain ring or Galois ring (Lee et al., 2012, Baek et al., 9 Apr 2026).

The construction has several variants, each governed by a specific algebraic input.

Setting	Required algebraic input	Length change
Binary $\F_2$	No extra hypothesis; $-1=1$ is automatically a square	$2n \to 2n+2$
Split $2n+2$0-ary $2n+2$1, $2n+2$2	Fix $2n+2$3 with $2n+2$4	$2n+2$5
Lee–Kim field/ring version	Units $2n+2$6 with $2n+2$7	$2n+2$8

For fields of characteristic not $2n+2$9, Kim’s original building-up construction already covered the cases $2n$0 a $2n$1-power or $2n$2, provided one can solve

$2n$3

in the alphabet. Lee–Kim completed the remaining field case $2n$4, then extended the same two-vector mechanism to $2n$5, Galois rings $2n$6, and arbitrary finite chain rings under the same existence condition on $2n$7 and $2n$8 (Lee et al., 2012).

2. Binary formulation

In the binary case, Kim’s method starts with a self-dual $2n$9 code $2n+4$0 with generator matrix

$2n+4$1

Choose a vector $2n+4$2 satisfying the Euclidean norm condition $2n+4$3, and define

$2n+4$4

Then the matrix

$2n+4$5

generates a binary self-dual code $2n+4$6. The construction is described as adding one “odd-weight” row to obtain a new self-dual code of length $2n+4$7 (Baek et al., 9 Apr 2026).

A worked example builds the unique binary self-dual $2n+4$8 code from the $2n+4$9 parent. The parent code $C \subset \F_q^n$0 is generated by $C \subset \F_q^n$1, which is self-dual because $C \subset \F_q^n$2. Choosing $C \subset \F_q^n$3 gives $C \subset \F_q^n$4 and

$C \subset \F_q^n$5

The extended generator matrix is

$C \subset \F_q^n$6

One checks directly that $C \subset \F_q^n$7 over $C \subset \F_q^n$8 and $C \subset \F_q^n$9, so $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$0 is the unique binary self-dual $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$1 code of minimum distance $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$2 (Baek et al., 9 Apr 2026).

3. Completion over odd finite fields

For fields of characteristic different from $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$3, the Lee–Kim construction uses two extension vectors. Let $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$4 be a self-dual $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$5 code over a field $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$6 with generator matrix $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$7 whose rows are $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$8. Assume there exist $C = C^\perp = \{\,v \in \F_q^n : (v,w)=0 \text{ for all } w \in C\},$9 such that

$(u,v)=\sum_{i=1}^n u_i v_i$0

Choose $(u,v)=\sum_{i=1}^n u_i v_i$1 satisfying

$(u,v)=\sum_{i=1}^n u_i v_i$2

For each $(u,v)=\sum_{i=1}^n u_i v_i$3, define

$(u,v)=\sum_{i=1}^n u_i v_i$4

and

$(u,v)=\sum_{i=1}^n u_i v_i$5

Then

$(u,v)=\sum_{i=1}^n u_i v_i$6

generates a self-dual $(u,v)=\sum_{i=1}^n u_i v_i$7 code over $(u,v)=\sum_{i=1}^n u_i v_i$8. The proof checks that all rows are mutually orthogonal under the standard inner product, that the dimension rises by exactly $(u,v)=\sum_{i=1}^n u_i v_i$9 to $\F_2$0, and hence that the resulting code is self-dual (Lee et al., 2012).

The open case in earlier work was $\F_2$1 with $\F_2$2. Lee–Kim resolve precisely that case by observing in Lemma 2.1 that there do exist $\F_2$3 with

$\F_2$4

Proposition 2.2 then shows that the same matrix construction works for any odd prime power $\F_2$5 when $\F_2$6 is even. Proposition 2.4 gives the converse: any self-dual $\F_2$7 code over $\F_2$8 with $\F_2$9 even and $-1=1$0 arises, up to coordinate permutation, from a self-dual $-1=1$1 code by the same building-up step (Lee et al., 2012).

4. Extension to $-1=1$2, Galois rings, and finite chain rings

Lee–Kim observe that the proof over fields uses only the bilinear form and the existence of units $-1=1$3 with $-1=1$4. This yields Proposition 3.1: if $-1=1$5 is a finite chain ring admitting units $-1=1$6 such that

$-1=1$7

and if $-1=1$8 is a self-dual code of even length $-1=1$9 over $2n \to 2n+2$0 with generator $2n \to 2n+2$1, then the same choice of $2n \to 2n+2$2, the same definitions of $2n \to 2n+2$3, and the same four-term correction vectors $2n \to 2n+2$4 produce a self-dual code of length $2n \to 2n+2$5 over $2n \to 2n+2$6 (Lee et al., 2012).

The converse is also stated in ring-theoretic form. Proposition 3.2 asserts that every self-dual code over $2n \to 2n+2$7 of length at least $2n \to 2n+2$8 and free rank at least $2n \to 2n+2$9 arises by this same building-up step. In particular, for $2n+2$00 or $2n+2$01 with $2n+2$02, $2n+2$03 odd, one first lifts $2n+2$04 from $2n+2$05 to $2n+2$06 by a Hensel-type lemma to obtain units satisfying $2n+2$07, and then the same construction applies (Lee et al., 2012).

For arbitrary finite chain rings, the ambient algebra is described through the unique maximal ideal $2n+2$08 with nilpotency index $2n+2$09. A linear code of length $2n+2$10 over such a ring has a generator matrix in a Smith-type block form,

$2n+2$11

The main chain-ring theorem states that if $2n+2$12 admits a solution of $2n+2$13 in $2n+2$14, then the two-vector building-up step extends every self-dual code of length $2n+2$15 to length $2n+2$16, and conversely every self-dual code with free part rank at least $2n+2$17 arises by iterating this step (Lee et al., 2012).

5. Algorithmic realization and representative constructions

The Lee–Kim procedure is explicitly algorithmic. Its input is a self-dual $2n+2$18 code $2n+2$19 over a ring or field admitting $2n+2$20, together with a generator matrix $2n+2$21 with rows $2n+2$22. One first precomputes any one pair $2n+2$23 solving $2n+2$24. One then enumerates candidates $2n+2$25 with $2n+2$26, and for each such $2n+2$27 enumerates $2n+2$28 such that $2n+2$29 and $2n+2$30. For each pair $2n+2$31, one computes $2n+2$32 and $2n+2$33, forms

$2n+2$34

builds the generator matrix $2n+2$35, lets $2n+2$36, tests whether $2n+2$37 is self-dual, records $2n+2$38 up to equivalence, and then may recurse to longer lengths or apply isomorphism checks to remove duplicates (Lee et al., 2012).

Several concrete families were obtained in this way.

Starting point	Output	Notable property
Pless symmetry $2n+2$39 code $2n+2$40 over $2n+2$41	945 new extremal $2n+2$42 codes	Each has trivial automorphism group
Length-$2n+2$43 code $2n+2$44 over $2n+2$45	Lengths $2n+2$46	Minimum Hamming weight $2n+2$47
Base codes over $2n+2$48	$2n+2$49 and $2n+2$50 families	New optimal or best known parameters

For ternary codes, $2n+2$51, so Lee–Kim take $2n+2$52 since $2n+2$53. Starting from the Pless symmetry $2n+2$54 code $2n+2$55, they choose $2n+2$56 of length $2n+2$57 with $2n+2$58, then $2n+2$59 with $2n+2$60 and $2n+2$61, and obtain $2n+2$62 codes. By judicious choice of $2n+2$63, they construct 945 new extremal self-dual ternary $2n+2$64 codes with trivial automorphism group. Combined with earlier 293 known codes, the total is at least 1238 inequivalent codes (Lee et al., 2012).

For $2n+2$65-codes, they start from the length-$2n+2$66 code $2n+2$67 generated by $2n+2$68 and $2n+2$69. Building up first yields a length-$2n+2$70 code $2n+2$71 with $2n+2$72. Iterating the construction gives length-$2n+2$73 codes, with eight inequivalent examples and $2n+2$74 up to $2n+2$75; length-$2n+2$76 codes, with 20 examples satisfying $2n+2$77; and length-$2n+2$78 codes, with 10 examples whose Construction $2n+2$79 lattices have distinct kissing numbers. These $2n+2$80-codes have minimum Hamming weight $2n+2$81, which is the best possible minimum Hamming weight that free self-dual codes over $2n+2$82 of these lengths can attain. By Construction $2n+2$83, they yield the unique optimal Type I lattices in dimensions $2n+2$84 and, at length $2n+2$85, the odd Leech lattice (Lee et al., 2012).

For $2n+2$86, Lee–Kim again use the $2n+2$87 case, with $2n+2$88 solving $2n+2$89, for example $2n+2$90, $2n+2$91. Starting from a length-$2n+2$92 base code $2n+2$93 from Gulliver–Harada, they build up to length $2n+2$94 and obtain at least 214 inequivalent $2n+2$95 codes, of which 207 are new, with automorphism group orders from $2n+2$96 to $2n+2$97. Starting from the length-$2n+2$98 code $2n+2$99, they build up to length $2n$00 and obtain 59 new self-dual $2n$01 codes with trivial automorphism group (Lee et al., 2012).

6. Isotropic lines, Hilbert symbols, and formalization

The 2026 formalization shows that Kim’s building-up construction of binary self-dual codes is equivalent to Chinburg–Zhang’s Hilbert-symbol construction. Chinburg–Zhang’s approach starts from the global cup-product pairing on the étale cohomology of the ring of $2n$02-integers in $2n$03, then uses a boxed normal form and a top-down Lagrangian reduction. The equivalence is expressed by the observation that deleting the distinguished hyperbolic block and the top row in the boxed form yields a self-orthogonal predecessor code, while re-inserting that hyperbolic pair by choosing an isotropic line recovers exactly Kim’s formulas $2n$04 and the appended row $2n$05 (Baek et al., 9 Apr 2026).

In this framework, the common algebraic input is the condition that $2n$06 be a square. In the binary case this is automatic because $2n$07. In the split $2n$08-ary case, one fixes $2n$09 such that $2n$10, which is equivalent to $2n$11 being a square and to the standard Euclidean plane over $2n$12 being hyperbolic. If $2n$13 is self-dual and $2n$14 satisfies $2n$15, then with

$2n$16

the new generator matrix is

$2n$17

The identity $2n$18 makes the $2n$19-plane spanned by $2n$20 hyperbolic and organizes the isotropic correction terms on that line (Baek et al., 9 Apr 2026).

The Lean 4 development formalizes the self-dual/Lagrangian interface, the hyperbolic split background, binary and $2n$21-ary building-up, boxed forms, and reverse reconstruction in a single file without any “sorry” placeholders. The paper states that all 256 key lemmas and theorems were formalized. As an application of the efficient form of generator matrices, it constructs optimal self-dual codes from the split boxed construction, including self-dual $2n$22 and $2n$23 codes over $2n$24, MDS self-dual $2n$25 and $2n$26 codes over $2n$27, and a self-dual $2n$28 code over $2n$29 (Baek et al., 9 Apr 2026).