In a large glass or plastic bowl, mix the sugar, salt, and diced chili in the lime juice. Season with pepper and cumin. Stir in the olive oil, garlic, onion, tomato, and cilantro. Gently mix in the salmon, then cover, and refrigerate for 4 hours.
To serve, drain the excess liquid from the salmon, then gently stir in the diced cucumber and if desired avocado chunks, and allow to rest at room temperature for 15 minutes.