The ${[46,9,20]_2}$ code is unique

TL;DR

This paper constructs a unique optimal binary linear code of parameters [46,9,20], proves its uniqueness and asymmetry, and establishes the non-existence of certain related codes, advancing the understanding of code optimality and uniqueness.

Contribution

It presents the first known [46,9,20]_2 code, proves its uniqueness and asymmetry, and shows non-existence of specific larger or related codes.

Findings

Constructed a [46,9,20]_2 code and proved its uniqueness.

Showed the code is asymmetric with a trivial automorphism group.

Established non-existence of [47,10,20]_2 and [85,9,40]_2 codes.

Abstract

The minimum distance of all binary linear codes with dimension at most eight is known. The smallest open case for dimension nine is length $n=46$ with known bounds $19\le d\le 20$. Here we present a $[46,9,20]_2$ code and show its uniqueness. Interestingly enough, this unique optimal code is asymmetric, i.e., it has a trivial automorphism group. Additionally, we show the non-existence of $[47,10,20]_2$ and $[85,9,40]_2$ codes.

Tables

Table 1. Table 1. Number of [ n , k , { 20 , 24 , 28 , 32 } ] 2 subscript 𝑛 𝑘 20 24 28 32 2 [n,k,\{20,24,28,32\}]_{2} codes.

k / n	20	24	28	30	32	34	35	36	37	38	39	40	41	42	43	44
1	1	1	1	0	1	0	0	0	0	0
2				1	1	2	0	3	0	3	0
3							1	1	2	4	6	9
4										1	4	13	26
5												3	15	163
6														24	3649
7															5	337794

Table 2. Table 2. Number of [ n , k , { 40 , 48 , 56 } ] 2 subscript 𝑛 𝑘 40 48 56 2 [n,k,\{40,48,56\}]_{2} codes.

k / n	40	48	56	60	64	68	70	72	74	75	76	77	78	79	80	81	82	83
1	1	1	1	0	0	0	0	0	0	0	0	0
2				1	1	2	0	2	0	0	2	0	0
3							1	1	2	0	3	0	5	0
4										1	1	2	3	6	10
5													1	3	11	16
6															2	8	106
7																	7	5613

Table 3. Table 3. Number of [ 84 , 8 , { 40 , 48 , 56 } ] 2 subscript 84 8 40 48 56 2 [84,8,\{40,48,56\}]_{2} codes per A 56 subscript 𝐴 56 A_{56} .

$A_{56}$	3	4	5	6	7	8	9
	25773	48792	26091	5198	450	17	1

Equations

$$n(k,d)\ge g(k,d):=\sum _{i=0}^{k-1}\lceil \frac{d}{2^i}\rceil ,$$

$$\left[n-w,k-1,\ge d-\lfloor w/2\rfloor \right].$$

$$\sum _{j=0}^n{K}_i(j)A_j(C)=2^k{B}_i(C),0\le i\le n$$

$$K_i(j)=\sum _{s=0}^n(-1{)}^s(\genfrac{}{}{0ex}{}{n-j}{i-s}\left)\right(\genfrac{}{}{0ex}{}{j}{s}),0\le i\le n$$

$$\alpha$$

$$\beta$$

$$\delta$$

$$T:=\sum _{i=0}^{\lfloor n/(2\Delta )\rfloor }{A}_{2\Delta i}$$

$$T=2^{k-a}-2^{k-a-t}$$

$$T=2^k-2^{k-a}+2^{k-a-t}$$

$$A_7+A_9+\sum _{i\ge 10}{A}_i=2^4-1=15\text{and}7A_7+9A_9+\sum _{i\ge 10}i{A}_i=2^{3}n=8n,$$

$$2A_9+3A_{10}+\sum _{i\ge 11}(i-7)A_i=8n-105.$$

$$\left(\begin{array}{c}1001010101110011011010001111001100100100000000\\ 1111100101010100100011010110011001100010000000\\ 1100110100001111101111000100000110101001000000\\ 0110101010010110101101110010100011001000100000\\ 0011101110101101100100101001010001011000010000\\ 0110011001111100011100011000110000111000001000\\ 0001111000011100000011111000001111111000000100\\ 0000000111111100000000000111111111111000000010\\ 0000000000000011111111111111111111111000000001\end{array}\right).$$

$$A_{20}+A_{24}+A_{28}+A_{30}+\sum _{i=1}^8{A}_{2i+30}$$

$$3A_{20}-A_{24}-5A_{28}-7A_{30}-\sum _{i=1}^8\left(2i+7\right)\cdot A_{2i+30}$$

$$5A_{20}+21A_{24}-27A_{28}-75A_{30}-\sum _{i=1}^8\left(8i^2+56i+75\right)\cdot A_{2i+30}$$

$$1x^0+(186+l)x^{40}+(69-2l)x^{48}+l{x}^{56}$$

Full text

Keywords: Binary linear codes, optimal codes

1. Introduction

An $[n,k,d]_{q}$-code is a $q$-ary linear code with length $n$, dimension $k$, and minimum Hamming distance $d$. Here we will only consider binary codes, so that we also speak of $[n,k,d]$-codes. Let $n(k,d)$ be the smallest integer $n$ for which an $[n,k,d]$-code exists. Due to Griesmer [7] we have

[TABLE]

where $\lceil{x}\rceil$ denotes the smallest integer $\geq x$. As shown by Baumert and McEliece [1] for every fixed dimension $k$ there exists an integer $D(k)$ such that $n(k,d)=g(k,d)$ for all $d\geq D(k)$, i.e., the determination of $n(k,d)$ is a finite problem for every fixed dimension $k$. For $k\leq 7$, the function $n(k,d)$ has been completely determined by Baumert and McEliece [1] and van Tilborg [12]. After a lot of work of different authors, the determination of $n(8,d)$ has been completed by Bouyukliev, Jaffe, and Vavrek [4]. For results on $n(9,d)$ we refer e.g. to [5] and the references therein. The smallest open case for dimension nine is length $n=46$ with known bounds $19\leq d\leq 20$. Here we present a $[46,9,20]_{2}$ code and show its uniqueness. Interestingly enough, this unique optimal code is asymmetric, i.e., it has a trivial automorphism group. Speaking of a $\Delta$-divisible code for codes whose weights of codewords all are divisible by $\Delta$, we can state that the optimal code is $4$-divisible. $4$-divisible codes are also called doubly-even and $2$-divisible codes are called even. Additionally, we show the non-existence of $[47,10,20]_{2}$ and $[85,9,40]_{2}$ codes. Our main tools – described in the next section – are the standard residual code argument (Proposition 2.2), the MacWilliams identities (Proposition 2.3), a result based on the weight distribution of Reed-Muller codes (Proposition 2.4), and the software packages Q-Extension [2], LinCode [8] to enumerate linear codes with a list of allowed weights. For an easy access to the known non-existence results for linear codes we have used the online database [6].

2. Basic tools

Definition 2.1**.**

Let $C$ be an $[n,k,d]$-code and $c\in C$ be a codeword of weight $w$. The restriction to the support of $c$ is called the residual code $\operatorname{Res}(C;c)$ of $C$ with respect to $c$. If only the weight $w$ is of importance, we will denote it by $\operatorname{Res}(C;w)$.

Proposition 2.2**.**

Let $C$ be an $[n,k,d]$-code. If $d>w/2$, then $\operatorname{Res}(C;w)$ has the parameters

[TABLE]

Some authors call the result for the special case $w=d$ the one-step Griesmer bound.

Proposition 2.3**.**

*([9], MacWilliams Identities)

Let $C$ be an $[n,k,d]$-code and $C^{\perp}$ be the dual code of $C$. Let $A_{i}(C)$ and $B_{i}(C)$ be the number of codewords of weight $i$ in $C$ and $C^{\perp}$, respectively. With this, we have*

[TABLE]

where

[TABLE]

are the binary Krawtchouk polynomials. We will simplify the notation to $A_{i}$ and $B_{i}$ whenever $C$ is clear from the context.

Whenever we speak of the first $l$ MacWilliams identities, we mean Equation (2) for $0\leq i\leq l-1$. Adding the non-negativity constraints $A_{i},B_{i}\geq 0$ we obtain a linear program where we can maximize or minimize certain quantities, which is called the linear programming method for linear codes. Adding additional equations or inequalities strengthens the formulation.

Proposition 2.4**.**

*([5, Proposition 5], cf. [10])

Let $C$ be an $[n,k,d]$-code with all weights divisible by $\Delta:=2^{a}$ and let $\left(A_{i}\right)_{i=0,1,\dots,n}$ be the weight distribution of $C$. Put*

[TABLE]

Then the integer

[TABLE]

satisfies the following conditions.

1. (1)

$T$* is divisible by $2^{\lfloor(k-1)/(a+1)\rfloor}$.* 2. (2)

If $T<2^{k-a}$, then

[TABLE]

for some integer $t$ satisfying $1\leq t\leq\max\{\alpha,\beta\}$. Moreover, if $t>\beta$, then $C$ has an $[n,k-a-2,\delta]$-subcode and if $t\leq\beta$, it has an $[n,k-a-t,\delta]$-subcode. 3. (3)

If $T>2^{k}-2^{k-a}$, then

[TABLE]

for some integer $t$ satisfying $0\leq t\leq\max\{\alpha,\beta\}$. Moreover, if $a=1$, then $C$ has an $[n,k-t,\delta]$-subcode. If $a>1$, then $C$ has an $[n,k-1,\delta]$-subcode unless $t=a+1\leq k-a-1$, in which case it has an $[n,k-2,\delta]$-subcode.

A special and well-known subcase is that the number of even weight codewords in a $[n,k]$ code is either $2^{k-1}$ or $2^{k}$.

3. Results

Lemma 3.1**.**

Each $[\leq 16,4,7]_{2}$ code contains a codeword of weight $8$.

• Proof. Let $C$ be an $[n,4,7]_{2}$ code with $n\leq 16$ and $A_{8}=0$. From the first two MacWilliams identities we conclude

[TABLE]

so that

[TABLE]

Thus, the number of even weight codewords is at most $8n/3-34$. Since at least half the codewords have to be of even weight, we obtain $n\geq\left\lceil 15.75\right\rceil=16$. In the remaining case $n=16$ we use the linear programming method with the first four MacWilliams identities, $A_{8}=0$, $B_{1}=0$, and the fact that there are exactly $8$ even weight codewords to conclude $A_{11}+\sum_{i\geq 13}A_{i}<1$, i.e., $A_{11}=0$ and $A_{i}=0$ for all $i\geq 13$. With this and rounding to integers we obtain the bounds $5\leq B_{2}\leq 6$, which then gives the unique solution $A_{7}=7$, $A_{9}=0$, $A_{10}=6$, and $A_{12}=1$. Computing the full dual weight distribution unveils $B_{15}=-2$, which is negative. $\square$

Lemma 3.2**.**

Each even $[46,9,20]_{2}$ code $C$ is isomorphic to a code with generator matrix

[TABLE]

• Proof. Applying Proposition 2.2 with $w=20$ on a $[45,9,20]$ code would give a $[25,8,10]$ code, which does not exist. Thus, $C$ has effective length $n=46$, i.e., $B_{1}=0$. Since no $[44,8,20]$ code exists, $C$ is projective, i.e., $B_{2}=0$. Since no $[24,8,9]$ code exists, Proposition 2.2 yields that $C$ cannot contain a codeword of weight $w=22$. Assume for a moment that $C$ contains a codeword $c_{26}$ of weight $w=26$ and let $R$ be the corresponding residual $[20,8,7]$ code. Let $c^{\prime}\neq c_{26}$ be another codeword of $C$ and $w^{\prime}$ and $w^{\prime\prime}$ be the weights of $c^{\prime}$ and $c^{\prime}+c_{26}$. Then the weight of the corresponding residual codeword is given by $(w^{\prime}+w^{\prime\prime}-26)/2$, so that weight $8$ is impossible in $R$ ($C$ does not contain a codeword of weight $22$). Since $R$ has to contain a $[\leq 16,4,7]_{2}$ subcode, Lemma 3.1 shows the non-existence of $R$, so that $A_{26}=0$. With this, the first three MacWilliams Identities are given by

[TABLE]

Minimizing $T=A_{0}+A_{20}+A_{24}+A_{28}+A_{32}+A_{36}+A_{40}+A_{44}$ gives $T\geq\tfrac{6712}{15}>384$, so that Proposition 2.4.(3) gives $T=512$, i.e., all weights are divisible by $4$. A further application of the linear programming method gives that $A_{36}+A_{40}+A_{44}\leq\left\lfloor\tfrac{9}{4}\right\rfloor=2$, so that $C$ has to contain a $[\leq 44,7,\{20,24,28,32\}]_{2}$ subcode. Next, we have used Q-Extension and LinCode to classify the $[n,k,\{20,24,28,32\}]_{2}$ codes for $k\leq 7$ and $n\leq 37+k$, see Table 1. Starting from the $337799$ doubly-even $[\leq 44,7,20]$ codes, Q-Extension and LinCode give 424207 doubly-even $[45,8,20]_{2}$ codes and no doubly-even $[44,8,20]_{2}$ code (as the maximum minimum distance of a $[44,8]_{2}$ code is $19$.) Indeed, a codeword of weight $36$ or $40$ can occur in a doubly-even $[45,8,20]_{2}$ code. We remark that largest occurring order of the automorphism group is $18$. Finally, an application of Q-Extension and LinCode on the 424207 doubly-even $[45,8,20]_{2}$ codes results in the unique code as stated. (Note that there may be also doubly-even $[45,8,20]_{2}$ codes with two or more codewords of a weight $w\geq 36$. However, these are not relevant for our conclusion.) $\square$

We remark that the code of Lemma 3.2 has a trivial automorphism group and weight enumerator $1x^{0}+235x^{20}+171x^{24}+97x^{28}+8x^{32}$, i.e., all weights are divisible by four. The dual minimum distance is $3$ ($A_{3}^{\perp}=1$, $A_{4}^{\perp}=276$), i.e., the code is projective. Since the Griesmer bound, see Inequality (1), gives a lower bound of $47$ for the length of a binary linear code with dimension $k=9$ and minimum distance $d\geq 21$, the code has the optimum minimum distance. The linear programming method could also be used to exclude the weights $w=40$ and $w=44$ directly (and to show $A_{36}\leq 2$). While the maximum distance $d=20$ was proven using the Griesmer bound directly, the $[46,9,20]_{2}$ code is not a Griesmer code, i.e., where Inequality (1) is satisfied with equality. For the latter codes the $2^{2}$-divisibility would follow from [13, Theorem 9] stating that for Griesmer codes over $\mathbb{F}_{p}$, where $p^{e}$ is a divisor of the minimum distance, all weights are divisible by $p^{e}$.

Theorem 3.3**.**

Each $[46,9,20]_{2}$ code $C$ is isomorphic to a code with the generator matrix given in Lemma 3.2.

• Proof. Let $C$ be a $[46,9,20]_{2}$ with generator matrix $G$ which is not even. Removing a column from $G$ and adding a parity check bit gives an even $[46,9,20]_{2}$ code. So, we start from the generator matrix of Lemma 3.2 and replace a column by all $2^{9}-1$ possible column vectors. Checking all $46\cdot 511$ cases gives either linear codes with a codeword of weight $19$ or the generator matrix of Lemma 3.2 again. $\square$

Lemma 3.4**.**

No $[47,10,20]_{2}$ code exists.

• Proof. Assume that $C$ is a $[47,10,20]_{2}$ code. Since no $[46,10,20]_{2}$ and no $[45,9,20]_{2}$ code exists, we have $B_{1}=0$ and $B_{2}=0$, respectively. Let $G$ be a systematic generator matrix of $C$. Since removing the $i$th unit vector and the corresponding column (with the $1$-entry) from $G$ gives a $[46,9,20]_{2}$ code, there are at least $1023$ codewords in $C$ whose weight is divisible by $4$. Thus, Proposition 2.4.(3) yields that $C$ is doubly-even. By Theorem 3.3 we have $A_{32}\geq 8$. Adding this extra inequality to the linear inequality system of the first four MacWilliams identities gives, after rounding down to integers, $A_{44}=0$, $A_{40}=0$, $A_{36}=0$, and $B_{3}=0$. (We could also conclude $B_{3}=0$ directly from the non-existence of a $[44,8,20]_{2}$-code.) The unique remaining weight enumerator is given by $1x^{0}+418x^{20}+318x^{24}+278x^{28}+9x^{32}$. Let $C$ be such a code and $C^{\prime}$ be the code generated by the nine codewords of weight $32$. We eventually add codewords from $C$ to $C^{\prime}$ till $C^{\prime}$ has dimension exactly nine and denote the corresponding code by $C^{\prime\prime}$. Now the existence of $C^{\prime\prime}$ contradicts Theorem 3.3. $\square$

So, the unique $[46,9,20]_{2}$ code is strongly optimal in the sense of [11, Definition 1], i.e., no $[n-1,k,d]_{2}$ and no $[n+1,k+1,d]_{2}$ code exists. The strongly optimal binary linear codes with dimension at most seven have been completely classified, except the $[56,7,26]_{2}$ codes, in [3]. The next open case is the existence question for a $[65,9,29]_{2}$ code, which is equivalent to the existence of a $[66,9,30]_{2}$ code. The technique of Lemma 3.2 to conclude the $4$-divisibility of an optimal even code can also be applied in further cases and we given an example for $[78,9,36]_{2}$ codes, whose existence is unknown.

Lemma 3.5**.**

Each $[\leq 33,5,15]_{2}$ code contains a codeword of weight $16$.

• Proof. We verify this statement computationally using Q-Extension and LinCode. $\square$

We remark that a direct proof is possible too. However, the one that we found is too involved to be presented here. Moreover, there are exactly $3$ $[\leq 32,4,15]_{2}$ codes without a codeword of weight $16$.

Lemma 3.6**.**

If an even $[78,9,36]_{2}$ code $C$ exists, then it has to be doubly-even.

• Proof. Since no $[77,9,36]_{2}$ and no $[76,8,36]_{2}$ code exists, we have $B_{1}=0$ and $B_{2}=0$. Proposition 2.2 yields that $C$ does not contain a codeword of weight $38$. Assume for a moment that $C$ contains a codeword $c_{42}$ of weight $w=42$ and let $R$ be the corresponding residual $[36,8,15]_{2}$ code. Let $c^{\prime}\neq c_{42}$ be another codeword of $C$ and $w^{\prime}$ and $w^{\prime\prime}$ be the weights of $c^{\prime}$ and $c^{\prime}+c_{42}$. Then the weight of the corresponding residual codeword is given by $(w^{\prime}+w^{\prime\prime}-42)/2$, so that weight $16$ is impossible in $R$ ($C$ does not contain a codeword of weight $38$). Since $R$ has to contain a $[\leq 33,5,15]_{2}$ subcode, Lemma 3.5 shows the non-existence of $R$, so that $A_{42}=0$. We use the linear programming method with the first four MacWilliams identities. Minimizing the number $T$ of doubly-even codewords gives $T\geq\tfrac{1976}{5}>384$, so that Proposition 2.4.(3) gives $T=512$, i.e., all weights are divisible by $4$. $\square$

Two cases where $8$-divisibility can be concluded for optimal even codes are given below.

Theorem 3.7**.**

No $[85,9,40]_{2}$ code exists.

• Proof. Assume that $C$ is a $[85,9,40]_{2}$ code. Since no $[84,9,40]_{2}$ and no $[83,8,40]_{2}$ code exists, we have $B_{1}=0$ and $B_{2}=0$, respectively. Considering the residual code, Proposition 2.2 yields that $C$ contains no codewords with weight $w\in\{42,44,46\}$. With this, we use the first four MacWilliams identities and minimize $T=A_{0}+\sum_{i=10}^{21}A_{4i}$. Since $T\geq 416>384$, so that Proposition 2.4.(3) gives $T=512$, all weights are divisible by $4$. Minimizing $T=A_{0}+\sum_{i=5}^{10}A_{8i}$ gives $T\geq 472>384$, so that Proposition 2.4.(3) gives $T=512$, i.e., all weights are divisible by $8$. The residual code of each codeword of weight $w$ is a projective $4$-divisible code of length $85-w$. Since no such codes of lengths $5$ and $13$ exist, $C$ does not contain codewords of weight $80$ or $72$, respectively.111We remark that a $4$-divisible non-projective binary linear code of length $13$ exists. The residual code $\hat{C}$ of a codeword of weight $64$ is a projective $4$-divisible $8$-dimensional code of length $21$. Note that $\hat{C}$ cannot contain a codeword of weight $20$ since no even code of length $1$ exists. Thus we have $A_{64}\leq 1$. Now we look at the two-dimensional subcodes of the unique codeword of weight $64$ and two other codewords. Denoting their weights by $a$, $b$, $c$ and the weight of the corresponding codeword in $\hat{C}$ by $w$ we use the notation $(a,b,c;w)$. W.l.o.g. we assume $a=64$, $b\leq c$ and obtain the following possibilities: $(64,40,40;8)$, $(64,40,48;12)$, $(64,40,56;16)$, and $(64,48,48;16)$. Note that $(64,48,56;20)$ and $(64,56,56;24)$ are impossible. By $x_{8}$, $x_{12}$, $x_{16}^{\prime}$, and $x_{16}^{\prime\prime}$ we denote the corresponding counts. Setting $x_{16}=x_{16}^{\prime}+x_{16}^{\prime\prime}$, we have that $x_{i}$ is the number of codewords of weight $i$ in $\hat{C}$. Assuming $A_{64}=1$ the unique (theoretically) possible weight enumerator is $1x^{0}+360x^{40}+138x^{48}+12x^{56}+1x^{64}$. Double-counting gives $A_{40}=360=2x_{8}+x_{12}+x_{16}^{\prime}$, $A_{48}=138=x_{12}+2x_{16}^{\prime\prime}$, and $A_{56}=12=x_{16}^{\prime}$. Solving this equation system gives $x_{12}=348-2x_{8}$ and $x_{16}=x_{8}-93$. Using the first four MacWilliams identities for $\hat{C}$ we obtain the unique solution $x_{8}={102}$, $x_{12}=144$, and $x_{16}=9$, so that $x_{16}^{\prime\prime}=9-12=-3$ is negative – contradiction. Thus, $A_{64}=0$ and the unique (theoretically) possible weight enumerator is given by $1x^{0}+361x^{40}+135x^{48}+15x^{56}$ ($B_{3}=60$). Using Q-Extension and LinCode we classify all $[n,k,\{40,48,56\}]_{2}$ codes for $k\leq 7$ and $n\leq 76+k$, see Table 2. For dimension $k=8$, there is no $[83,8,\{40,48,56\}]_{2}$ code and exactly 106322 $[84,8,\{40,48,56\}]_{2}$ codes. The latter codes have weight enumerators

[TABLE]

($B_{2}=l-3$), where $3\leq l\leq 9$. The corresponding counts are given in Table 3. Since the next step would need a huge amount of computation time we derive some extra information on a $[84,8,\{40,48,56\}]_{2}$-subcode of $C$. Each of the $15$ codewords of weight $56$ of $C$ hits $56$ of the columns of a generator matrix of $C$, so that there exists a column which is hit by at most $\left\lfloor 56\cdot 15/85\right\rfloor=9$ such codewords. Thus, by shortening of $C$ we obtain a $[84,8,\{40,48,56\}]_{2}$-subcode with at least $15-9=6$ codewords of weight $56$. Extending the corresponding $5666$ cases with Q-Extension and LinCode results in no $[85,9,\{40,48,56\}]_{2}$ code. (Each extension took between a few minutes and a few hours.) $\square$

Lemma 3.8**.**

Each $[\leq 47,4,23]_{2}$ code satisfies $A_{24}+A_{25}+A_{26}\geq 1$.

• Proof. We verify this statement computationally using Q-Extension and LinCode. $\square$

We remark that there a 1 $[44,3,23]_{2}$, 3 $[45,3,23]_{2}$, and 9 $[46,3,23]_{2}$ codes without codewords of a weight in $\{24,25,26\}$.

Lemma 3.9**.**

Each even $[\leq 46,5,22]_{2}$ code contains a codeword of weight $24$..

• Proof. We verify this statement computationally using Q-Extension and and LinCode. $\square$

We remark that there a 2 $[44,4,22]_{2}$ and 6 $[45,4,22]_{2}$ codes that are even and do not contain a codeword of weight $24$.

Theorem 3.10**.**

If an even $[117,9,56]_{2}$ code $C$ exist, then the weights of all codewords are divisible by $8$.

• Proof. From the known non-existence results we conclude $B_{1}=0$ and $C$ does not contain codewords with a weight in $\{58,60,62\}$. If $C$ would contain a codeword of weight $66$ then its corresponding residual code $R$ is a $[51,8,23]_{2}$ code without codewords with a weight in $\{24,25,26\}$, which contradicts Lemma 3.8. Thus, $A_{66}=0$. Minimizing the number $T_{4}$ of doubly-even codewords using the first four MacWilliams identities gives $T_{4}\geq\frac{2916}{7}>384$, so that Proposition 2.4.(3) gives $T_{4}=512$, i.e., all weights are divisible by $4$. If $C$ contains no codeword of weight $68$, then the number $T_{8}$ of codewords whose weight is divisible by $8$ is at least $475.86>448$, so that Proposition 2.4.(3) gives $T_{8}=512$, i.e., all weights are divisible by $8$. So, let us assume that $C$ contains a codeword of weight $68$ and consider the corresponding residual $[49,8,22]_{2}$ code $R$. Note that $R$ is even and does not contain a codeword of weight $24$, which contradicts Lemma 3.9. Thus, all weights are divisible by $8$. $\square$

Proposition 3.11**.**

If an even $[118,10,56]_{2}$ code exist, then its weight enumerator is either $1x^{0}+719x^{56}+218x^{64}+85x^{72}+1x^{80}$ or $1x^{0}+720x^{56}+215x^{64}+88x^{72}$.

• Proof. Assume that $C$ is an even $[118,10,56]_{2}$ code. Since no $[117,10,56]_{2}$ and no $[116,9,56]_{2}$ code exists we have $B_{1}=0$ and $B_{2}=0$, respectively. Using the known upper bounds on the minimum distance for $9$-dimensional codes we can conclude that no codeword as a weight $w\in\{58,60,62,66,68,70\}$. Maximizing $T=\sum_{i}A_{4i}$ gives $T\geq 1011.2>768$, so that $C$ is $4$-divisible, see Proposition 2.4.(3). Maximizing $T=\sum_{i}A_{8i}$ gives $T\geq 1019.2>768$, so that $C$ is $8$-divisible, Proposition 2.4.(3). Maximizing $A_{i}$ for $i\in\{88,96,104,112\}$ gives a value strictly less than $1$, so that the only non-zero weights can be $56$, $64$, $72$, and $80$. Maximizing $A_{80}$ gives an upper bound of $\frac{3}{2}$, so that $A_{80}=1$ or $A_{80}=0$. The remaining values are then uniquely determined by the first four MacWilliams identities. $\square$

The exhaustive enumeration of all $[117,9,\{56,64,72\}]_{2}$ codes remains a computational challenge. While we have constructed a few thousand non-isomorphic $[115,7,\{56,64,72\}]_{2}$ codes, we still do not know whether a $[117,9,56]_{2}$ code exists.