Convolutional neural network for sparse one-hot representation
I would not apply convolutional neural networks to your problem (at least from what I can gather from the description).

Convolutional nets' strengths and weaknesses are related to a core assumption in the model class: Translating patterns of features in a regular way either has a minor impact on the outcome, or has a specific useful meaning. So a pattern 1 0 1 seen in features 9,10,11 is similar in some way to the same pattern seen in features 15,16,17. Having this assumption built in to the model allows you to train a network with far fewer free parameters when dealing with e.g. image data, where this is a key property of data captured by scanners and cameras. 

With one-hot encoding of features, you assign a feature vector index from a value or category essentially at random (via some hashing function). There is no meaning to translations between indices of the feature vectors. The patterns 0 0 1 0 1 0 0  and 0 0 0 1 0 1 0 can represent entirely different things, and any associations between them are purely by chance. You can treat a sparse one-hot encoding as an image if you wish, but there is no good reason to do so, and models that assume translations can be made whilst preserving meaning will not do well.

For such a small sparse feature vector, assuming you want to try a neural network model, use a simple fully-connected network.