Errata to Complete Caps in Projective Space which are Disjoint from a Subspace of Codimension Two

David L. Wehlau

The published version of Complete Caps in Projective Space which are Disjoint from a Subspace of Codimension Two contains an error in the proof of Theorem 4.1. This error affects Proposition 7.1 and Proposition 8.1.

The correct statement and proof of Theorem 4.1 are as follows:

Theorem 4.1 Suppose that A and B are such that Equation 1.3 is satisfied for all i = 1, 2 …, 2^{n−r−1}. Suppose 0 < t + u < 2^{n−r−1}, that t ≠ u ,

and that either \overline{A_{f}} or \overline{B_{f}} is non-periodic . Further suppose that Ĉ \ C ⊆

∪_{i=1}^{2^{n−r−1}} (A(i) + B(i)). Then S is a complete cap in Σ.

Proof By construction, S is a cap in Σ. Also Equation 1.3 guarantees that every point of A′ and every point of B′ lies on at least one secant of S through a point of C. Thus we need to show that every point of H_{∞} and every point of H_{C} \ C lies on a secant of S.

Without loss of generality suppose that \overline{A_{f}} is non-periodic. Consider the

cap \overline{S_{1}} in \overline{Σ} given by \overline{S_{1}} = \overline{A_{f}} ⊔ \overline{B_{ne}} ⊔ \{\overline{C}\}. Applying Theorem 3.5 we see that for every point \overline{c_{k}} of \overline{H_{C}} different from \overline{C} there exists i ≤ t and j ≥ t + 1 such that \overline{a_{i}} + \overline{b_{j}} = \overline{c_{k}}. Since the coset H_{A}(i) is entirely contained in S, and H_{B}(j)∩S ≠ ∅, this means that every point of the coset H_{C}(k) = H_{A}(i)+H_{B}(j) lies on a secant to S.

By assumption, every point of Ĉ \ C is contained in A + B.

Hence it only remains to prove that every point of H_{∞} lies on at least one secant to S. Note that every point of F lies in H_{A}(1) ⊕ H_{A}(1) (or in H_{B}(1) ⊕ H_{B}(1) if t = 0). Thus we consider a point z ∈ H_{∞} \ F and write \overline{z} for the point of \overline{Σ} corresponding to the coset of F generated by z. As above, if \overline{z} ∈ \overline{A_{f}} + \overline{A_{ne}} or \overline{z} ∈ \overline{B_{f}} + \overline{B_{ne}} then z lies on a secant line to S. Thus we assume, by way of contradiction, that \overline{z} + \overline{A_{f}} ⊂ \overline{A_{e}} and \overline{z} + \overline{B_{f}} ⊂ \overline{B_{e}}. The first inclusion implies that t ≤ u while the second implies that u ≤ t. Thus t = u contradicting our hypothesis.

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2 David L. Wehlau

Here are the corrected versions of Remark 4.3 and Remark 4.4.

Remark 4.3 Note that, if t or u is odd, then \overline{A_{f}} or \overline{B_{f}} respectively is nonperiodic. Thus for a fixed value of t + u with 1 ≤ t + u ≤ 2^{n−r−1} − 1 we may take t = 1, for example, to arrange that \overline{A_{f}} is non-periodic. Furthermore, if t + u ≠ 2 we may simultaneously arrange that t ≠ u.

Remark 4.4 .4 There do exist complete caps for which t + u = 0 (and thus \overline{A_{f}} = \overline{B_{f}} = ∅). See Example 5.4. The condition that t ≠ u is also sufficient but not necessary.

The corrected version of paragraph 4 of Section 5, as well as two new paragraphs which should follow paragraph 4 are give below:

Then |S| = (2^{r} − 1) + 2s + 2^{r}(2^{n−r−1} − s) = 2^{n−1} + 2^{r} − 1 − (2^{r} − 2)s. By Remark 4.2, s cannot equal 0. Furthermore by Theorem 4.1 (and Remark 4.3) there do exist complete caps of this form for all s = 1, 2, …, 2^{n−r−1} − 3 and

s = 2^{n−r−1} − 1. Thus we find complete caps of this form of all cardinali-

ties: 2^{n−r} + k(2^{r} − 2) + 1 for k = 2, 3, …, 2^{n−r−1} − 2 and k = 2^{n−r−1} where

2 ≤ r ≤ n − 2.

Next we consider the case s = 2^{n−r−1} − 2. Then u + v = 2 and either u = v or else both \overline{A_{f}} and \overline{B_{f}} are periodic. Reordering the cosets we may suppose |H_{A}(i)| = |H_{B}(i)| = 1 for i = 3, 4, …, 2^{n−r−1}. Write {α_{i}} = H_{A}(i) and {β_{i}} = H_{B}(i) for i = 3, 4, …, 2^{n−r−1}. We consider first the case that t = u = 1. Examining the proof of Theorem 4.1 we see that every point of ℙ𝔾(n, 2) lies on a secant to S except possibly the 2^{r} points in the coset H_{A}(1) + H_{A}(2) = H_{B}(1) + H_{B}(2) = {z_{1}, z_{2}, …, z_{2^{r}}} ⊂ H_{∞}. These points z_{k} can only lie on secants of the form α_{i} + α_{j} = z_{k} or β_{i} + β_{j} = z_{k}. Since β_{i} +β_{j} = α_{i} +α_{j} we may restrict our attention to the α_{i}. Thus we see that S is complete if and only if for every k = 1, 2, …, 2^{r} there exists 3 ≤ i(k) < j(k) ≤ 2^{n−r−1} such that α_{i(k)} + α_{j(k)} = z_{k}. Since the α_{i} all lie in different cosets, it is clear that α_{i(1)}, α_{j(1)}, …, α_{i(2^{r})}, α_{j(2^{r})} must be distinct. Thus we must have 2(2^{r}) ≤ 2^{n−r−1} − 2 or equivalently 2r + 3 ≤ n. The case (t, u) = (2, 0) (or (0, 2)) is entirely similar to the above case. The only difference being that the 2^{r} points which may not lie on secants to S form a coset in H_{C} instead of in H_{∞}. In summary we see that there exists a complete cap S with |Ĉ \C| = 1 and with |S| = 2^{n−r} +2^{r+1} +2^{r} −5 (and s = 2^{n−r−1} −2) if and only if n ≥ 2r + 3.

Errata to Complete Caps in Projective Space which are Disjoint from a Subspace of Codimension Two 3

Here is the corrected version of Proposition 7.1:

Proposition 7.1 Let S be a complete cap in ℙ𝔾(n, 2) with n ≥ 4. Further suppose that S meets a hyperplane K_{C} of ℙ𝔾(n, 2) in exactly 3 points. Then |S| is one of the numbers: 2^{n−2} + 5, 2^{n−2} + 9, 2^{n−2} + 11, 2^{n−2} + 13, …, 2^{n−1} + 1 or, if n ≥ 7, |S| may also be 2^{n−2} + 3 or 2^{n−2} + 7. Moreover complete caps of all these sizes exist for all corresponding values of n.

Here is the corrected version of Proposition 8.1

Proposition 8.1 Let S be a complete cap in ℙ𝔾(n, 2) with n ≥ 5. Further suppose that S meets a hyperplane K_{C} of ℙ𝔾(n, 2) in exactly 4 points. Then 2^{n−2} + 8 ≤ |S| ≤ 2^{n−1} − 2 or else |S| = 2^{n−1} + 1. Moreover except for n = 6 for sizes 29 and 30, complete caps of all these sizes

and this structure exist for all n ≥ 5.

September 1, 20001

David L. Wehlau

Department of Mathematics and Computer Science

Royal Military College

Kingston, Ontario

Canada K7K 7B4

  wehlau@rmc.ca

I also thank Alexander Davydov for suggesting I clarify the statement of Proposition 8.1 (by adding and this structure ) and for pointing out an error on page 355 line -11 (=last sentence on page 7 of the corrected version). The valid values for k are k = 2, 4, 5, …, 2^{n−r−1} where 2 ≤ r ≤ n − 2.

September 15, 2003