A CONJECTURE ON N1L CONFIGURATIONS

N1L configurations occur in linear double error correcting codes. Bounding their length is a question of interest. A conjecture is given which would yield a lower bound. AMS Subject Classification: 94B05


Introduction
As in [1], an N 1L configuration is a set S of weight 5 vectors, where there is a weight 5 "anchor" vector v and a position i ∈ v, such that for w ∈ S − {v}, i ∈ w and |v ∩ w| = 2; and further the linear span has minimum weight 5.
An N ′ 1L configuration is a configuration of weight 3 vectors, which may be obtained from an N 1L configuration by deleting v, and the positions of v in the remaining vectors.Together with the incidence matrix M , a partition of the rows into 4 parts must be given.Note that in each part, the rows are disjoint (i.e., the column sets of the 1 entries are).
Let r denote the number of rows and c the number of columns in M .The minimum value of c for a given value of r is a value of interest.Result of an exhaustive search [1] showed that for r = 4, 8, 12 the minimum value of c is 8,12,16 respectively.A further search showed that for r = 16 there are configurations with c = 19.
Here the output of these computations is examined further, and a conjecture made.Some partial results and computational results are given.
In an array, number rows from 0 at the top, and columns from 0 on the left.The weight |v| of a 0-1 vector v is the number of 1's.

TM Configurations
Let s ≥ 3 be an integer, intended to be the part size of an N 1L configuration with constant part size.A T matrix is defined to be a s × 4 matrix of 0's and 1's, where each row or column has at most one 1, and there are three 1's altogether.A TM configuration is a matrix M comprised of a 4 × (s − 2) array of T matrices, with the following properties.
1. Let T ij be the T matrix at position ij in the array thereof.Let z ij be the position of the 0 column of T ij .Then for fixed j the four z ij are distinct.
In particular each column of M has weight 3. 2. If 3 ≤ s ≤ 6 each row has weight 1 or 2, and if 6 ≤ s each row has weight 2 or 3.
Conjecture 1.For r = 4s where s ≥ 3 there are N 1L configurations M which are the concatenation of a TM configuration M 1 and a second matrix M 2 .
Note that the total weight of M 2 is 24.In particular, if the conjecture is true c ≤ r + 16 may be achieved.The computations of [1] produce examples verifying the conjecture for s = 3, 4.

Proof of Conjecture 1 for s ≤ 6
Define a two-part restriction to be a restriction on an incidence matrix M , which is necessary for M to be an N ′ 1L configuration.The following two restrictions are such: 1.If r 1 , r 2 are rows from different parts them Proof.If s ≤ 6 then every row of M 2 is nonempty, so it suffices to verify that no linear combination v of at most 4 rows of the N 1L configuration has 0 < |v| < 5.The linear combinations may be assumed to involve at least 2 parts.Let w 0 denote the weight in the anchor columns, and w 2 the weight in the M 2 columns.
The following cases suffice, where the case label gives the number of rows per part.
Proof.By theorem 2 it suffices to show that there is a TM configuration satisfying restrictions 1 and 2. Here, we merely note that such are readily found by a computer program.

Further Computations
The computations of [1] produce examples for s = 3, 4 where M 2 may be taken as having 12 weight 2 columns, and c = r + 4 may be achieved.For s = 4 there is an M 2 with 11 columns, and c = r + 3 may be achieved.
Searches over TM configurations can be conducted.This provides exact values for s = 3, 4 for TM based N 1L configurations, and examples for s = 5.
Define a T − matrix to be an s × 3 matrix of weight 3 with disjoint rows and columns, i.e., a T matrix with the 0 column removed.A T − A matrix is defined to be the concatenation of s − 2 T − matrices, which satisfies requirement 2 for a TM configuration.These can be canonicalized up to row permutation.(This and other canonicalizations descried here are readily accomplished using the Nauty [2] library).
For a selection of 4 T − A's consider the TM configuration, where in part i the T − 's of the T − A matrix have a 0 column inserted at position i.The resulting TM matrix need only be considered if it satisfies the two-part restrictions 1 and 2 given above.The remaining configurations can be canonicalized, where the row permutations act on the parts.
Given such a possible M 1 , a search can be conducted through an additional 12 columns, looking for N ′ 1L configurations.When s = 4 there are 536 TM matrices of interest.For 1 of them, the minimum length of an M 2 is 19, and there are M 2 's with 10 weight 2 columns and a weight 4 column.For an additional 5, the minimum length of an M 2 is 12, and there are M 2 's with 12 weight 2 columns.
When s = 5 there are 4919449 TM matrices of interest.Let M 1 be a random TM configuration.Starting with a trivial M 2 , weight 1 columns may be combined randomly until the resulting row space is no longer weight 5.This random search finds an N ′ 1L configuration with r = 20 and c = 24 quite rapidly.For a remark on using Nauty, the graphs which need to be canonicalized are bipartite.After canonicalization the n 1 × n 2 submatrix which is the 0-1 matrix where the i, j entry is 1 if there is an edge between vertex i of part 1 and vertex j of part 2 may be extracted.The entire matrix may be reconstructed from this.In the case s = 5 above, where n 1 = 20 and n 2 = 16, the size was reduced from 288 bytes to 40 bytes, allowing the canonicalized graphs to fit in memory.The time required to generate the TM's was 325 minutes.
r 11 , r 12 are rows from part 1, and r 21 , r 22 are rows from part 2, then |(r 11 ∪ r 12 ) ∩ (r 21 ∪ r 22 )| ≤ 3. Say that an extension M of a TM configuration to an N ′ 1L configuration is trivial if M 2 consists of 24 weight 1 columns.Note that for such, if M 1 satisfies the above two restrictions then M does.Theorem 2. If M 1 is a TM configuration satisfying restrictions 1 and 2 above, and s ≤ 6, then a trivial extension M of M 1 is an N ′ 1L configuration.